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This happened yesterday:

Then this:

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:

Piketty Simulator

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 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 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 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 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 and gs 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.


Piketty says something in a way that sounds like he takes it, as so many do, as axiomatic:

“…growth always includes a purely demographic component and a purely economic component, and only the latter allows for an improvement in the standard of living.”

The nit I’d like to pick with this received trusim is marginal relative to its broad accuracy, but is still worth noting – there are economies of scale to absolute population. These manifest in two interrelated ways – consumption and investment choices that are only “unlocked,” if it were, when total population crosses certain thresholds, and future per-capita-growth that results from past choices that were contingent upon absolute population.

I can illustrate these by giving three examples – one purely the former, one purely the latter, one a mix.

A purely “unlocked” choice would be a more specialized service that could not achieve scale relative to fixed costs without a large enough absolute population given a fixed share of population interested in the service. Think “shop that only sells customized meeples” or more conventionally “Latverian restaurant.” This doesn’t affect to the level of per-capita income or output, now or in the future, but improves living standards by providing a greater diversity of quality consumption options.

A purely future-oriented choice might be an aircraft carrier. Today, nobody benefits. But in the long term, if an aircraft carrier in the most optimistic framing maintains peace, security, and a stable order, this allows for greater per-capita growth (and fewer destabilizing interruptions) in the future, though in the present it registers as output that brings little utility to the public at large. Obviously two things must be noted –  military investment does not always increase peace, security, and stability; and even assuming it does, there are many, much more cynical, interpretations of how military power projection leads to future per-capita growth for the projectors.

A mixed choice would be cet bruyant objet du désir, a large subway system. It both increases consumption options and quality available to present individuals – lots of people prefer riding trains! – while also being an investment that increases long-term per-capita-growth rates.

This is not the most important point in the world, but since Piketty made it I found it a good time to quibble with it.


Miles Kimball and Yichuan Wang find that high government debt doesn’t cause low GDP growth, and Kimball says he finds that surprising, as does Matt Yglesias. But as I suggested in a post last month, I’m not really surprised by this at all.

Governments tax or borrow. The former is withdrawing money from the economy in exchange for nothing (or perhaps a promise not to sanction the taxed) while the latter withdraws money from the economy in exchange for a piece of paper. That’s debt! Evil, evil, debt! Oh, no!

Wait, let’s start over.

The goverment decides it wants to do something it isn’t already doing, and therefore needs to command a higher share of total social production going forward than it has been. Developed-world governments don’t directly commandeer social resources, they claim through the proxy of money, by spending it. Assuming an economy at full capacity (whatever that means), if the government commandeers resources by spending money without removing any money from the economy then you’d have inflation, unless the central bank raises interest rates substantially, which would likely have undesireable negative effects. So the government attempts to roughly balance the resources claims it makes using money by withdrawing an equivalent amount of money from society. Sometimes it does this through taxes, which has some desireable properties (no future obligations on the state, can be used Pigovianly) and some undesirable ones (unintended consequences, involuntary, discourages desireable activity). Borrowing also has some desireable properties (voluntary, compensates those who part with their money) and some undesireable properties (obligates the state).

Therefore, there are two key intertwined questions to be asked about this new government activity, which remember is centrally about taking some resources deployed previously to some private purpose and redirecting them to some other, presumably public purpose – is the new activity more valuable than the activity(s) it is supplanting, and how is it being financed? They are intertwined because the latter question informs the former.

Let’s say we all agree that this new government project – let’s say it’s a SUPERTRAIN, for fun – is widely considered to be of higher value than the marginal private activity it supplants regardless of how it is funded. The government could raise taxes to fund it, but unless it is taxing something undesireable (like carbon or booze or Kardashians) this would have the drawback of incurring some "deadweight loss," not to mention other unintended consequences. It could also borrow the money, which would have two consequences. Firstly, it would supplant something different – rather than raising the cost of work or carbon emissions, it would be more likely to supplant a capital investment of some kind somewhere in the economy. Secondly, it would obligate the government.

And to what would it obligate the government? Key to understanding this is that governments, unlike Lannisters, never pay their debts. They cleverly disguise this fact by paying their debts in full and on time. Huh? From the perspective of a borrower, you get your interest payments, and then your principal in full. But from the perspective of a government, you don’t pay the principal back out of tax revenue, you pay it by rolling over the debt and issuing new debt in the amount of the principal. This works because of NGDP growth (both the RGDP growth and inflationary components). In fact, we’re still likely rolling over all the debt we incurred from WWII, which back then was 110% of NGDP but today is less than 2% of NGDP.

So really what the government does when it issues a bond is issue itself a negative perpetuity. And the key to understanding the value of a perpetuity is knowing the interest rate, since the PV = C/r. Therefore, the obligation on the government is much more dependent on the interest rate path than on the nominal coupon value.

But that interest rate path isn’t just some made-up thing – it’s fundamentally related to NGDP growth. Don’t believe me? Here’s the fed funds rate divided by the NGDP growth rate:

Inline image 1

So when recessions happen, the ratio spikes (and whether it spikes up or down is very interesting), but otherwise it’s very steady; if you exclude just the 12 of 223 periods where the absolute value of the ratio is greater than 3, you get an average of 0.8 and a standard deviation of 0.6.

So what does that mean? As interest rates grow, so does the obligation on the government – but it also implies that the government’s ability to meet that obligation is growing in tandem. Which suggests that, while governments cannot borrow limitlessly, the pain point at which government indebtedness begins to inflict structural economic harm is vastly higher than previous assumptions.

Japan, for example, is often cited as an example of government debt creating a huge drag/time bomb/giant vengeful lizard that is harming Japan’s economy. But since 1990, Japan’s debt/GDP ratio increased from 67% to 211% and GDP-per-capita…grew! Significantly! Not awesomely, not enough to catch-up with the US (in fact, it fell behind), but grow it did. Certainly more than you might think it would if the 90% monster were real and starting smashing major cities or something.

Many people have begun to worry whether the seemingly-inevitable Japanese debt crisis is nearing as yield have crept up. But yields have crept up because NGDP-growth-expectations have crept up. As long as they increase in tandem, contra Noah Smith, Japan should always be able to pay its debts.

And I’d be willing to put money/my reputation on this point. While Noah Smith is 100% right that bets != beliefs, I am nonetheless willing to agree in principle to any reasonably-valued bet that neither Japan nor the United States will default over any arbitrary time period. Any takers?

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