This happened yesterday:

Fun fact: r > g is not a sufficient condition for explaining rising inequality.

— Josh Hendrickson (@RebelEconProf) April 10, 2014

Then this:

@RebelEconProf …go on.

— Squarely Rooted (@squarelyrooted) April 10, 2014

And you can read the rest of the conversation from there (it was actually quite civil), but for the purposes of this post, it brought me back to the Piketty Simulator I ginned up a little while back to test Piketty’s second law, and I expanded it. And what do you know – Hendrickson is roughly 50% right. And figuring out exactly why gets at the heart of Piketty’s project. Check it out:

So if you open up the spreadsheet and play with it yourself – and you should! spreadsheets are fun! – you should know a few things. Firstly, continuing my stated opposition to grecoscriptocracy, I have changed Piketty’s alpha and beta, the capital share of national income and the long-run equilibrium capital/income ratio, to the Hebrew *aleph* (א) and *bet *(ב). I have also created a new variable of interest, which I assigned the Hebrew *gimel *(ג), which we’ll get to a bit later.

In the spreadsheet, you can set initial conditions of the following five variables – the initial levels of capital and national income, and Piketty’s *r, s, *and *g* – the return on capital, the savings rate, and the growth rate. The spreadsheet then tells you a few things, both over the course of three centuries (!) and the long-term equilibrium.

Firstly, it tells you א and ב. Secondly, assuming invariant wealth shares, it tells you the share of national income that goes to the “rentier class” given any given wealth share.

The other thing it tells you, which is key to the first part of this discussion, is ג, which can be best defined as the capital perpetuation rate; it is the percentage of the “*r*” produced by capital that needs to be saved in order to maintain the existing ב. It can be defined, and derived, in two ways. The first is *g*/*r, *which is intuitive; it can also be derived as *s*/א, which may be less intuitive, but it also really important. Because it shows both why Hendrickson is wrong and why he was right.

The key to Hendrickson’s point is that *s *is really important to the inequality path. Which is correct! But the other point is that inequality can, and will, rise regardless of *s** *so long as *r>g* and Piketty’s big assumption is true. More on the latter later, but play with both math the simulator first.

The math first - *s*/א is a clear way to derive ג: it’s the ratio of the share of national income devoted to capital formation divided by the share of national income produced by existing capital. But if you decompose it (fun with algebra and spreadsheets in one post – I’m really hitting a home run here) you’ll see that since א=*r**ב and ב=*s*/*g* then you’ve got *s *in the numerator and the denominator and it cancels out. That’s why I put both derivations of ג - ג and ג prime – in the spreadsheet; even though one is directly derive from the savings rate, you can change *s* all you want and ג remains stubbornly in place. *Other* things change, but not ג.

This is important because it decomposes exactly what Piketty is getting at with his *r*>*g* inequality. Essentially, there are two different things going on. One is the perpetuability of capital, the other is the constraint on capital-driven inequality. As you change *s *in the spreadsheet, you’ll see that the rentier share of national income changes accordingly as the long-run ב increases; you’ll also notice that “rentier disposable income” changes accordingly. Hey, what’s that? It’s the amount of income leftover to rentiers after they’ve not only not touched the principal but also *reinvested to keep pace with growth*.

And indeed, you’ll see if you change *r *and *g* that as they get closer and closer, regardless of how large the capital to income ratio is the rentiers need to plow more and more of their returns from capital into new investment to ensure their fortunes keep pace with the economy. Indeed, if *r*=*g*, then rentiers must reinvest 100% of their capital income or else inexorably fall behind the growth of the economy as a whole.

In summary, Piketty’s *r*>*g* is telling us whether the owners of substantial fortunes – think of them as “houses,” not individual people – can maintain or improve their privileged position relative to society as a whole *ad infinitum*. Given *r *and *g*, *s* tells us how privileged that position really is. Even with a 50% savings rate (!), if *g* = 4% while *r* = 5% then even though a rentier class that owns 90% (!) of national capital captures 56% of national income, they can only dispose of just over 11% of national income or else they will be slowly but surely swallowed into the masses. On the other hand, if *s* = 6%, fairly paltry, but *g* is only 1% relative to *r*‘s 5%, then rentiers only capture, initially, 22.5% of national income; but they can *spend* 18% and still maintain their position; if they spend just the 11% above, they can start increasing their already very privileged position (though this model doesn’t account for that).*

So Hendrickson is both right – you need to incorporate *s* to compute the long-run inequality equilibrium, while also wrong in that, so long as we’re not yet at that equilibrium, *r>g* can and, at the very least likely if not necessarily inevitably, produce rising inequality. So while the share of national income that goes to creating new capital limits the ability of capitalists to increase their capital income to the point where it truly dominates society, so long as *r > g*, they not only need never fear of losing their position, but also through careful wealth management and, defined very relatively, frugality, expand it over time, at least until they hit the limit defined by *s**.*

But therein lies the rub. All these simulations, which echo Piketty’s work**, operate from a central fundamental assumption that, if altered, can topple the entire model (both Piketty’s and mine) – that *r, s, *and* g* are exogenous and independent. Now, Piketty himself doesn’t exactly claim that, but he does claim (both in *Capital* and in some of his previous, more technical economic work) that both theoretically there are many compelling models in which they largely move independently, especially within “reasonable” ranges, and in practice these values have been fairly steady over time and that changes in their medium-to-long-term averages, to the extent they are interconnected, have sufficiently low elasticities that, for example, *r* (and therefore א) decline slower than ב increases and therefore the dominance of capital increases. He derives this a little more technically in his appendix on pgs. 37-39, and discusses it in his book around pages 200-220; you can also check out this working paper to show how a production function with a constant elasticity of substitution > 1 can not only theoretically produce a model consonant with his projections but also matches the trend in Western countries over the past few decades.

These assumptions in many ways cut deeply and sharply against a lot of different assumptions, theories, and models about the economy that many people hold to, advocate for, and have a great deal of influence. And demonstrating conclusively or empirically how related they are can be maddeningly circular and also ripe territory for statistical arcana that most people don’t understand and, as Russ Roberts has pointedly noted, even those who do don’t really find convincing. But fundamentally, if you believe that *r*, *g*, and *s* are sufficiently independent and exogenous, you can view income distribution as a largely zero-sum game set by systems that states can to a substantial degree alter without changing those values; but if you view them as connected in vital feedback loops, you may be loathe to tax *r* for fear of depressing *s* and thereby depress *g*; your game is negative sum, not zero. How you view this bedrock question, a question hard to resolve conclusively through either theory or empirics, is going to determine a lot of what you take away from *Captial*.

*I’d love to create a model that shows variant rentier shares of national wealth and national income over time, but that’s not for this post, at least.

**One thing Piketty doesn’t stress but this spreadsheet makes clear is just how long the processes Piketty describes take to play out. Given the default society I plugged into the spreadsheet – *r*=5%, *s*=12%, *g*=1%, C=3, NI=1 – a rentier class that own 90% of total wealth, while projected to capture over half of national income in the long run, only captures ~14% initially; after 50 years, it is still capturing less than 30% of national income; and even after two centuries, it is still 6% of national income short of it’s long-run equilibrium, which is quite a bit. Obviously expecting fundamental aspects of society to be invariant for that long in our post-industrial world is probably very unrealistic, but it gives you a sense of the scale of the dynamics this book is grappling with.