Matt Breunig has posted three excellent new pieces on Thomas Piketty’s analysis of the dynamics of inequality. They’re at the Demos Policyshop blog, and can be found here, here and here. Since Breunig comes to many of the same interpretive conclusions I have reached myself, I will just refer the reader to these pieces without much further comment on them. But I do want to call special attention to one thing Breunig says in response to a recent critical essay on Piketty by John Aziz. In characterizing Piketty’s account of inequality, Aziz says that according to Piketty inequality will tend to increase when the rate of economic growth for the entire economy is less than the average return on capital. But Breunig notes in response:

*This is fine enough as a gloss of an explanation, but is it not strictly true. Piketty’s actual point *[is]* that the larger the **spread** between r and g, the more forcefully the dynamics of capital income **pushes** in the direction of increasing wealth inequality.*

Breunig raises an important point here, and it bears emphasis. Piketty couches most of his arguments about equality and inequality in terms of *forces of divergence* and *forces of convergence*. His approach is to identify those conditions under which the forces of divergence will predominate, and if so, how strongly they will predominate. His view is that at the beginning of the 21^{st} century, the conditions appear to be in place for the forces of divergence to acquire renewed strength, although he also stresses that nothing is certain, and the exact course of 21^{st} century inequality depends on a host of political, demographic, technological and economic factors. One thing Piketty routinely stresses, however, is that the forces for divergence operate very strongly when r is “significantly and durably” higher than g, and automatically lead to a very high concentration of wealth.

I want to add a bit of precision to these initial statements, and in the process shed some light on why Piketty lays very significant stress on the fact that different kinds of wealth owners earn different rates of return on their wealth, and also on the fact that the wealthy save their incomes at higher rates than those who are less wealthy. Both of these phenomena play an important role in Piketty’s analysis of the forces for divergence and the structure of inequality in Chapters 10, 11 and 12 of *Capital in the Twenty-First Century*.

To help fix ideas and focus on the most important phenomena, let’s suppose that our society contains a class of *pure rentiers*: these are people who perform no paid labor, and the entirety of whose income is therefore income from capital. It does not matter for the arguments that follow exactly how large we imagine that class of pure rentiers to be. Here are four questions we can ask about these rentiers and their role in the dynamics of inequality:

Q1: Under what conditions will the pure rentiers’ share of national *income* increase? In other words, under what conditions will the pure rentiers’ income rise faster than the rate of growth of national income?

Q2: How *rapidly* does the pure rentier share of income increase in those conditions when it is increasing?

Q3: Under what conditions will the pure rentiers’ share of national *wealth* increase? In other words, under what conditions will pure rentier wealth rise faster than national wealth?

Q4: How *rapidly* does the pure rentier share of wealth increase in those conditions when it is increasing?

Let’s take these questions up one at a time. In what follows, let s and r be the national savings rate and average rate of return on capital respectively. Let ρ be the pure rentiers’ average savings rate, and let r_{R} be the pure rentiers’ average rate of return on their wealth. Since it is possible for the values of these quantities to vary over time, we could attach subscripts to the variables to index them to a year. But since the year will always be clear from context in what follows, we can dispense with the subscripts. We will also use the expressions “YR_{i}” and “WR_{i}” to refer to the rentiers’ income in year i and the rentiers’ wealth in year i, respectively, while “Y_{i}” and “W_{i}” will refer to the values of national wealth and national income for year i. Also, for ease of expression, I will usually drop the terminology “pure rentiers” in favor of the simpler “rentiers.”

So let’s turn to our first question: Under what conditions will rentier income rise faster than the rate of growth of national income? Note first that national wealth grows according to the following rule:

- W
_{i+1 }= W_{i}+ sY_{i}

while rentiers’ wealth grows according to the corresponding rule:

- WR
_{i+1 }= WR_{i}+ ρYR_{i}

Since all rentier income is capital income then we have the following two equations:

- YR
_{i }= r_{R}(WR_{i}) - YR
_{i+1 }= r_{R}(WR_{i+1})

From 2 and 4, we can conclude:

- YR
_{i+1 }= r_{R}(WR_{i}+ ρYR_{i})

The rate at which rentiers’ income grows is (YR_{i+1}/YR_{i}) – 1, and from 3 and 5 we arrive with the help of some algebra at:

- YR
_{i+1}/YR_{i}– 1 = ρ(YR_{i}/WR_{i}) = ρr_{R}

So, then, rentiers’ income grows faster than national income when ρr_{R} > g, or alternatively, when ρ > g/r_{R}. (I argued in an earlier piece that the relevant condition was ρ > g/r, but that was under the simplifying assumption that all wealth owners had the same rate of return.)

We can also note that it is very easy to prove that rentier wealth grows at the same rate as rentier income, since rentier wealth comes entirely from capital income. Thus the rentier rate of wealth increase is also ρr_{R}. (I will leave that proof to the reader.) So we can now state the answer to our first question:

**A1: The rentier share of national income will increase just in case ρr _{R} > g.**

Let’s turn now to our second question which concerns the *rate* of this increase. This question is fairly easy to answer, now that we have the answer to our first question in hand. The rate at which the rentier share of income grows is (YR_{i+1}/Y_{i+1})/(YR_{i}/Y_{i}) – 1. From statement 6, we get:

- (YR
_{i+1}/Y_{i+1})/(YR_{i}/Y_{i}) – 1 = [(1+ρr_{R})YR_{i}/(1+g)Y_{i}]/(YR_{i}/Y_{i}) – 1

which then reduces to:

- (YR
_{i+1}/Y_{i+1})/(YR_{i}/Y_{i}) – 1 = (1+ρr_{R})/(1+g) – 1

And so that gives us the answer to our second question:

**A2: The rate at which the pure rentier share of income increases is (1+ρr _{R})/(1+g) – 1.**

Now for the third question: Under what conditions does the rentier share of wealth increase? Clearly the rentiers’ *share* of wealth increases if the rate of rentier wealth increase ρr_{R} exceeds the rate of national wealth increase. The rate of national wealth increase is (W_{i+1}/W_{i}) – 1. By statement 1, above, we have:

- (W
_{i+1}/W_{i}) – 1 = [(W_{i}+ sY_{i})/W_{i}] – 1

And by elementary algebra, we then get:

- (W
_{i+1}/W_{i})- 1 = [1 + s(Y_{i}/W_{i})] – 1 - (W
_{i+1}/W_{i}) – 1 = s(Y_{i}/W_{i})

Using Piketty’s notion for the capital-to-income ratio β_{i} = W_{i}/Y_{i}, we can conclude from 11 that:

- (W
_{i+1}/W_{i})- 1 = s/β_{i}

So our conclusion is that the rentier share of wealth increases when ρr_{R} > s/β_{i}. Alternatively, we can express this condition as β_{i} > s/ρr_{R}. And so that gives us the answer to our third question:

**A3: The pure rentier share of wealth increases just in case β _{i} > s/ρr_{R}.**

Notice that it is possible to have conditions under which rentier income and rentier wealth are growing faster than national income, while they are at the same time *not* growing faster than national wealth, and where the rentier share of wealth is actually decreasing. For example, suppose r and r_{R }are both 5%, while the rate of growth of national income is 1%, the rate of national savings in 9% and rate of rentier savings is 30%. Then the rate of rentier wealth increase and rentier income increase are both equal to ρr= 1.5%, and β must exceed s/ρr = 6 in order for the rentier share of wealth to increase. Suppose however that β is only 5.75. Then β will slowly rise toward 9 in accordance with Piketty’s Second Fundamental law β → s/g. The rentier share of wealth will decrease until β exceeds 6, and will then reverse direction and increase indefinitely.

It should be clear that the fact that rentiers save at a higher rate than everybody else is quite important to the dynamics of inequality. The greater the rentier savings rate ρ, the faster rentier income and wealth will grow, and the lower β has to be for the rentier share of wealth to begin its increase. If however, to take an example, rentiers received the same 5% average rate of return on capital as everyone else, *and* saved at the same rate 9% as everyone else, then the capital-to-income ratio would have to be greater than 20 for the rentier share of wealth to be increasing.

Piketty calls attention to the role of differences in savings rates among different classes of individuals in several places, predominantly in his discussion of the role of inherited wealth. (See, for example, the discussions on p. 351 and pp. 399-401.) I have simplified some of these ideas by looking at a separate class of pure rentiers rather than flows of inherited wealth, but I will present Piketty’s inherited wealth framework in a later post.

But note at this point that while it is important for rentiers to save at higher rates than the national average in order for their share of wealth to increase, it is by no means necessary that they save 100% of their capital income, or anywhere near 100% of that income, for wealth divergence to occur. Savings rates around 25% to 35% will do the trick in most circumstances. This latter point stands opposed to claims to the contrary by Lawrence Summers and other commentators on Piketty, who have argued that Piketty presupposes 100% re-investment of capital income by the wealthy in showing how income and wealth inequality increase.

Finally, let us turn to our fourth question. What can we say about the *rate* at which the rentier share of wealth is increasing in those circumstances in which it is increasing? For any year i, the rentier share of wealth H_{i} is defined:

- H
_{i}=_{df}WR_{i}/W_{i}

And the rate of increase in the rentier share of wealth, Πi, is given by:

- Π
_{i}=_{df}(H_{i+i}/H_{i}) – 1

From 13 and 14, we derive:

- Π
_{i}= (WR_{i+1}/W_{i+1})/(WR_{i}/W_{i}) – 1

And from statements 1 and 2, we then get:

- Π
_{i}= [(WR_{i}+ ρYR_{i})/(W_{i}+ sY_{i})]/(WR_{i}/W_{i}) – 1

From statement 3 and the definition of the capital-to-income ratio β, we obtain:

- Π
_{i}= [(WR_{i}+ ρr_{R}WR_{i})/(W_{i}+ sW_{i}/β_{i})]/(WR_{i}/W_{i}) – 1

And with just a little bit more algebraic manipulation, we reach:

- Π
_{i}= (1+ ρr_{R})/(1+ s/β_{i}) – 1

And so this gives us the answer to our fourth question:

**A4: The rate at which the rentier wealth share rises is (1+ ρr _{R})/(1+ s/β_{i}) – 1.**

Now what can we conclude about the pace of rentiers’ wealth share increase just by looking at this formula? One thing we can say is that, other things being equal, the rentiers’ share of wealth will increase faster if they have a higher rate of savings ρ or a higher rate of return on their wealth r_{R}. We can also see that as β_{i} increases, the quantities s/β_{i} and 1 + s/β_{i} decrease, and as a result the rate at which the rentiers’ wealth share increases will rise. This is another route to validating Piketty’s assertion that the higher the capital-to-income ratio goes, the stronger are the forces of divergence. It also justifies the attention Piketty gives to the higher rates of return on capital that the world’s wealthiest people receive compared to the rates others receive. This is a central theme in Chapter 12 of Capital in the Twenty-First Century, a chapter devoted to the dynamics of *international* inequality. Piketty says there:

*It is easy to see that such a mechanism can automatically lead to a radical divergence in the distribution of capital. If the fortunes of the top decile or top centile of the global wealth hierarchy grow faster for structural reasons than the fortunes of the lower deciles, then inequality of wealth will of course tend to increase without limit. This inegalitarian process may take on unprecedented proportions in the new global economy. In view of the law of compound interest discussed in Chapter 1, it is also clear that this mechanism can account for very rapid divergence, so that if there is nothing to counteract it, very large fortunes can attain extreme levels within a few decades. Thus unequal returns on capital are a force for divergence that significantly amplifies and aggravates the effects of the inequality r > g. Indeed, the difference r − g can be high for large fortunes without necessarily being high for the economy as a whole. *(p. 431)

Let’s put some specific numbers into the formula appearing in A4 to develop some intuitions about the ways in which the rentier share of wealth might evolve under varying circumstances. First, suppose we have a society in which the capital-to-income ratio is 6, the rentiers’ return on capital 5%, the national savings rate is 10% and the rentiers’ savings rate is 30%. Then the rentiers’ share of wealth would actually be decreasing at a small annual rate of – 0.16% which would be sufficient to produce a 10-year decrease in the rentiers’ wealth share of – 1.63%, if that annual rate remained consistent. However, it will *not* remain consistent. The capital-to-income ratio β will rise toward 10 in accordance with the rule β → s/g, and once it reaches s/ρr_{R}, which in this case is equal to 6.67, then in accordance with A3, the rentiers’ share of wealth will again begin to rise, and the rate at which it increases will rise along with β.

Suppose now, instead, that we have a society in which the capital to income ratio is 8 and the rentiers’ return on capital 7%, while the national savings rate is still 10% and the rentiers’ savings rate is still 30%. Then the rentiers’ share of wealth would grow at an annual rate of 0.84% which would be sufficient to produce a 10-year increase in the rentiers’ wealth share of 8.72%, if the annual rate again remained constant. But it will not remain consistent, because once again β will be rising, and as it rises the annual *rate* at which rentiers’s wealth share increases will also increase.

Notice, however, that even in the first scenario, where there is an initial period during which the rentiers’ wealth share is actually decreasing, the rentiers’ share of *income* would grow quite healthily during the same period if the growth rate of national income were sufficiently low. With a 1% annual growth rate, the rentiers’ income would grow at 1.5% per year and their share of income would grow at a ½ of 1% per year. Here we see an illustration of Piketty’s point that low growth provides an environment hospitable to the growth of inequality.

Two final points must be made about all of these calculations and ruminations. In describing these dynamic phenomena, I have been treating the rentiers’ rate of savings and rentiers’ return on capital as constants. But in the real world, as Piketty emphasizes, that is not likely to be the case. The richer people get, the larger the proportion of their income they can save and reinvest. And the very wealthiest individuals are able to achieve large economies of scale, and employ highly proficient full-time investment experts, in seeking out and obtaining higher rates of return on their investments. Thus there is a additional accelerating dynamic to the growth of wealth inequality that the formulas in this piece don’t capture.

The second point that must be made is that, of course, these kinds of increase can’t be sustained indefinitely. *Eventually*, the rates of return on capital for both rentiers and everybody else are likely to fall. *Eventually*, the capital-to-income ratio may stabilize.* Eventually*, the rich stop increasing their savings rates along with wealth. But as Piketty emphasizes, before that happens the wealthy may succeed in cementing a highly inegalitarian social structure into place. It will be no comfort to most people in society to realize that wealth and income inequality have stopped growing and have achieved a steady state if that steady state has already become a neo-feudal nightmare.

I would suggest setting up a stock-flow consistent model and try modelling the flows. You (and Piketty) set the national savings independently of the savings of the sectors. I am unsure whether that would work.

The other potential issue is that we are concerned about the wealth of individual households, not the “pure rentier” sector as a whole. Fortunes are dissipated by being split amongst multiple heirs, which was not the case a long time ago. Additionally, new fortunes are often created at the expense of old ones. These factors reduce inequality, and estimated attrition rates from these factors would need to be added into your models.

I’d agree that inequality is rising. But the assumption that equity returns going forward will be comparable to historical returns looks like a stretch.

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