Sunday, October 19, 2014

INEQUALITY DEATH SPIRAL!!!... misses the point

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.


  1. I think you've copied something down wrong, if β = s/g, then if g=0 then B=infinity. Still, the points you made in the text seem reasonable.

    1. Long run relations, exogenous parameters. These are also net returns and savings net of depreciation (which he makes clear in the text). Positive savings net of depreciation and zero growth will drive beta to infinity. The trouble is with the exogenous parameters, not the equation.

      Of course in the book he also talks at length about the determinants of the parameters too - they certainly aren't assumed to be fixed.

  2. Hold on a second Daniel. Are you saying the following two propositions?

    (1) Even though lots of people--fans and foes alike--think "r>g" is important to Piketty's thesis, Piketty himself doesn't give it central stage in his case.

    (2) Piketty isn't really making a case that wealth inequality will go up; instead he's talking about capital income inequality.

    1. 1. It's absolutely center stage. I didn't suggest otherwise. It is not INEQUALITY DEATH SPIRAL!

      2. Of course he's making the case that wealth inequality will go up.

    2. OK, so are you saying Piketty is NOT arguing that wealth will become alarmingly concentrated over the coming decades, unless governments take action?

    3. Of course he is arguing that. Can I ask where these questions are originating Bob?

    4. Perhaps this will help Bob - my whole point here is that what Piketty definitely does not say is "as long as r > g, inequality will always be increasing".

      r > g is most directly concerned with the capital share and you can have r > g with a stable capital share. It is indirectly related to inequality insofar as capital incomes drive inequality, with high returns on capital an important, but there are many more intervening forces and the relationship is not as direct as on the capital share side (which is why all these relations were introduced in the part of the book about the capital share).

  3. I like this post overall, but I worry that you're erring too much toward believing Piketty thinks β = s/g should be interpreted as an equality. As Current points out, this leads to obvious absurdities: the main thing the equation is supposed to show is the direction in which the capital/income ratio will increase. With this in mind, the interpretation of the 'β = infinity' result is that β will increase indefinitely for any positive savings rate if g = 0, which is more reasonable. Generally, Piketty is not especially interested in the hypothetical steady-state equilibrium implied by his model - he's looking to describe changes.

    1. I disagree. It's a long run relation to be sure but he definitely thinks it's an equality.

      The absurd results come out of absurd parameters, not an absurd relation.

      Certainly what we live through is the transition dynamics. But think back to the parts of the book where he describes the steady capital-income ratios, the interwar shocks, and his expectation that it'll get back to seven I believe it was. It is definitely an equality. (The equal sign is a nice tip-off on that one).

    2. OK well when I say it isn't an equality, what I mean is that in reality the equality will never hold. It would be better put as β → s/g. I think Piketty puts it as he does just to make it more accessible, and he does refer to it as an 'asymptotic law' immediately after introducing it.

      Circumstances change too often and the process of convergence happens far too slowly for the equality ever to hold. To steal a numerical example from someone else:

      "So, for example, if growth dropped to 1/8th of one percent, the initial capital-to income ratio were 5/1, and national savings remained constant at 10%, then β would begin to grow at about 1.9% per year, and grow at continually decreasing rates after that. It would in fact take 33 years for β to exceed 8/1. If g dropped all the way to zero, then the initial growth rate is simply s/β, which in this case is 2%. And it would still take 33 years to exceed 8/1."

      Do you really think Piketty believes we will actually ever reach a steady state? To me it seems the 'transition period' just refers to the period before the capital/income ratio starts to rise due to inherited wealth, rather than the period before β = s/g. His whole focus is on changes and dynamics over equilibrium.

    3. Like I said, the equality holds as a long run relation. It does not hold as a non-long-run relation.

      α = r*β of course does hold in the short run.

      re: "Do you really think Piketty believes we will actually ever reach a steady state?"

      Sure - part of the point of the capital-income ratio discussion was that it was steady for a long period of time before the wars and that it's on it's way to converging towards the long run equilibrium again. Of course the precise ratio is going to move around a little but precision really is too much to hope for in this case.

      But I thought he said pretty clearly that we are heading towards β = 7 or so, just as it had stabilized there previously.

      I may not be sure exactly what you're asking for. If you simply mean that over the 21st century it will jump between 6.85, 7.1, 7.06872, 6.98, etc. then sure. It's going to move because the parameters move. I don't know what amplitude to expect but the analysis suggests (and Piketty seems to expect) that it's going to converge back up to something like 7 (I think - I don't have book on my but that's what I recall).

    4. I think to get anywhere we'd have to start quoting Piketty directly, which probably isn't worth the time, but I will say that I think his projected rough capital income ratio is not based on entering certain values into β = s/g but on previously observed historical values for β. If you think there's something in Piketty which directly implies he thinks the steady state is practically relevant then I'll be happy to reread that section.

    5. Before we move on to economics let's look at the mathematical problem, you mention:
      * Long run relations.
      * exogenous parameters.
      * These are net returns and savings net of depreciation. (Why is depreciation netted from savings?)

      None of these things can change how formulas work. An equation that gives a ratio can't have a pole in it, if it does there's a mistake somewhere. I can understand why someone would argue that if g=0 then β will tend to some theoretical maximum, I think that's wrong but that's an economics argument. However, that maximum can only be *1* not infinity because it's a ratio.

      I haven't read Piketty's book, but I think you're misinterpreting this part of it. Of course if Piketty doesn't really use this result as Unlearningecon says then the issue is mostly academic.

    6. Sorry I was thinking in terms of the fundamental equation of the Solow model where to get capital accumulation you have to subtract out depreciated capital from the added savings.

      I'm not proposing any of it changes how formulas work. There is no hole or mistake as far as I'm aware. What mistake do you think there is? What is the long run equilibrium of an economy where growth is zero but savings are positive? The capital-income ratio goes to infinity. Where is the hole?

      If this were an identity that holds true at every point in time (like the equation for alpha) this would be a problem, but as far as I know it's not.

      re: "However, that maximum can only be *1* not infinity because it's a ratio."

      That makes no sense at all. Why can it only be 1 as a maximum? Even with the positive g's of the real world the capital-income ratio is on the order of six or seven.

    7. This isn't a Piketty thing it's a Solow thing. It's been around for sixty years. And actually rearranging the terms it's been around since the 30s. None of the equations here are new to Piketty's book.

    8. I see what you mean about the capital ratio. I'm still not sure about it though. Krussell and Smith say the problem is with depreciation. The proper equation isn't s/g it's s/(g-d) where d is the ratio of capital that degrades per year. I think it make much more sense if these equations are given explicitly as time series in the form e.g. β(t+1)=s(t)/(g(t)-d(t)).

      I still don't get why saving makes β tend towards infinity though.

    9. Krussell and Smith's problem is addressed on page 178 - like I said above, these are all net of depreciation. I definitely fault Piketty for simplifying the equations and burying the details further back in the book. For 90% of readers it probably achieves his aim and makes it easier. For ten percent, if they don't read all the details further back in the book carefully, it has created a lot of issues. It's Piketty's book though so that's certainly on Piketty.

      The problem with infinity only comes in with your scenario of zero long run growth. Beta is the capital-income ratio. If the bottom never grows and the top grows as a result of savings every period your capital-income ratio is going to go towards infinity. The logic of the equation is fine as far as I can tell, but the circumstance described by those parameters seems unrealistic.

  4. OK Daniel let's try it like this: Here's Kevin Drum's assessment of what Piketty believes:

    "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."

    Now at first, it seemed in this blog post that you had a problem with that.

    But now, after interrogating you, it seems like you agree with every word in that statement.

    So...what is your actual problem with Drum? Even though you agree with 100% of what he said, you are worried that some people might think Drum was saying something that Drum really wasn't saying?

    1. I am honestly not clear on what's unclear here - I was hoping it was clear that this is the part of the statement that I have an issue with: "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". I included the first half of the quote so people had context for the "But..."

      Is that clearer?

    2. OK, we're almost there Daniel... :)

      Drum is saying, "I know r>g doesn't imply that the rich will necessarily get richer. In fact, empirically that hasn't been the case in the last few centuries. So liberals who love Piketty and think he's on to something need to answer a question: Why should r>b all of sudden push up wealth concentration going forward, when it hasn't done so for the last few centuries? I've looked at the reasons Piketty gives to explain why it should suddenly have that effect, and don't find his reasons plausible."

      So what is wrong with Drum's post? He certainly isn't saying "r>g should drive wealth concentration to infinity," he's saying the opposite.

    3. Oh, and Drum *also* is not saying, "That fool Piketty thinks r>g necessarily drives all wealth into the hands of the top 0.01%."

      No, Drum is accurately reporting Piketty's position--that r>g will be a strong force to explain the huge jump in wealth concentration in the coming decades, absent government action--and Drum is then saying he thinks Piketty has a weak case.

      So, what problem do you have with this? As far as I can tell, you agree with every element of Drum's post.

    4. So Drum says liberals have a nice gotcha but they have a problem. We've had r > g for centuries but inequality hasn't been increasing for centuries.

      I'm saying this is a misuse of r > g. r > g is associated with a stable capital share of income and a stable capital to income ratio. Inequality is only indirectly related to r > g and we have no nice formula saying r > g does anything in particular to have it steadily increasing. Certainly we have scenarios where we think it could contribute to increasing inequality.

      I am simply making the point that Debraj Ray has been making that r > g is an important but standard result and it's not really the most interesting part of the story. Drum is playing into critics' narratives when he overemphasizes it like this.

    5. I read Drum as saying "Piketty isn't saying what the IGM is saying but there are still major problems with his argument"

      I am saying "You are misusing r > g, Drum".

      You seem to be thinking Drum is using the stability of inequality to make my point, but I see him using it to say at several points that Piketty may be wrong about r > g.

  5. On top of this we know a couple things:

    1. That g is falling simply as a result of the demographic transition,

    There are other factors, including: a slowdown in effective technological progress (the low hanging fruit has been picked); and, NIMBY is holding up productivity enhancing investments.


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