I was disheartened to read Kevin Drum

write this about the IGM Piketty question today:

"Piketty doesn't say that *r > g* has been a big driver of income inequality in recent years. He says only that he thinks it will be a big driver *in the future.*

This is good clean fun as a gotcha. But liberals should understand that it also exposes one of the biggest weaknesses of Piketty's argument: *r > g* has been true for centuries, but the rich have not gotten steadily richer over that time. Wealth concentration has stayed roughly the same."

It's nice that it's dawning on more and more people that the IGM survey is not that consequential for Piketty, and it's nice that Drum is spreading that point. But I think he's adding to the confusion by perpetuating this idea that if

*r > g* it means that inequality grows without bound. This narrative is part of what's driving peoples' incredulity about an inequality which (

as Debraj Ray has pointed out very clearly) is a standard result that pops out of standard growth models and doesn't have any of the dire inequality death-spiral implications people impute to it.

Piketty doesn't use it to predict a death spiral either. It's one of many fundamental equations he uses to talk about the capital-income ratio and the capital income as share of total income ratio. Rather than obsess over the

*r > g* inequality death spiral stuff, I think people need to spend a little time with this set of equations (and I think guys like Kevin Drum should take the opportunity to walk readers through it). If you made it through middle school, you can handle the math.

First, the share of income from capital of national income (

*α*) is:

*α = r*β*
Where r is the rate of return on capital and

*β* is the capital-income ratio (the ratio of the capital stock to annual income). This is straightforward, I hope. The capital-income ratio already has national income as its denominator, and the rate of return to capital times the amount of capital you have earning that return is capital income.

Next we have the equation governing

*β*, the capital-income ratio:

*β = s/g*
*g* is the growth rate of the economy (national income), and

*s* is the savings rate. Economies that save a lot but don't grow fast will eventually accumulate a large capital stock relative to national income. Of course this can be substituted in the first equation so that we've got both

*β *and

*α *written in terms of the other three variables:

*α = (r*s)/g*
So whatever

*r*,

*g*, and

*s* are even when

*r > g,* it's important to realize that the capital share of income is going to stabilize at some point. Piketty says this can happen in a number of ways. The classic answer in economics is that

*r* is going to decline (Piketty refers to this as "too much capital kills the return to capital"). It's a diminishing returns story. You could imagine ways that

*s* and

*g* could adjust too either for endogenous reasons (in other words - there's something about the way the economy works that pushes these variables in a certain direction) or for socio-political or technological reasons. And it's likely to be in flux in the real world and across societies of course. But one thing is very clear:

*r > g* does not tell us that capitalists' share just keeps increasing forever and ever - certainly it doesn't tell us that that's a necessity.

Part of the Piketty story is that we are in a period of convergence which is why things are changing now. The wars and the depression wiped out significant portions of the capital stock, and post-war institution subdued convergence back to these steady state values for

*β *and

*α*. As some of these institutions have been dismantled, convergence seems to be happening more rapidly. On top of this we know a couple things:

1. That

*g* is falling simply as a result of the demographic transition, and since

* g* is in the denominator of both

*β *and

*α*, these quantities get less stable as growth slows.

2. Wage income has been an important factor driving inequality in the late twentieth century, and as generations turn over that wage income becomes a new avenue for the fundamental equations above to kick in and begin converging again.

None of this is a death spiral story - it's more of a convergence story. Piketty suggests we might not like what we're converging too and we might want to consider institutional arrangements that might push things in a nicer direction.

Before closing I'd also emphasize a point I've made before on Piketty - the fundamental equations are really about capital income and the capital share,

__not__ wealth or income inequality directly. Capital income is a big piece of the puzzle of income inequality particularly because of what we know about the ownership of capital, but inequality is much more institutionally determined than capital dynamics.