Notes and Finger Exercises on Thomas Piketty’s “Capital in the Twenty-First Century”: The Honest Broker for the Week of April 19, 2014

There Are Four r’s

When I look at Thomas Piketty’s big book, I see one thing that he failed to do that I think he really should have done. A large part of the book is about the contrast between “r”, the rate of return on wealth, and “g” the growth rate of the economy. However, there are four different r’s. And in his book he failed to distinguish between them.

The four different r’s are:

  1. The real interest rate at which metropolitan governments can borrow: call this r1.

  2. The real interest rate that is the actual average return on wealthin the society and economy: call this r2.

  3. The real interest rate that is the average risky net rate of accumulation–what capital receives, minus the risk of confiscation or destruction or taxation, plus appreciation in valuation multiples, minus what is spent in order to keep the world in the appropriate social position: call this r3.

  4. A measure of the extent to which capital and wealth serve as an effective claim on income independent of how much capital there is–a standardized measure of what the society and economy’s return on wealth would be at some standardized ratio of wealth to annual income: say, 4: call this ρ.

These four r’s are very different animals.

The first r, r1, is what Larry Summers is talking about when he talks about secular stagnation. When that r1 falls to a level equal to minus the rate of inflation, the economy is in big trouble. At that point, wealthholders would rather become coupon-clipping rentiers holding government bonds then invest in industry of any sort. Full employment can then be attained only via:

  1. A bubble that produces unrealistic and unsustainable expectations of the profits from investing in industry.

  2. The government borrowing money and buying stuff on a large scale.

  3. A higher rate of trend inflation that relaxes the zero lower bound constraint on safe government debt interest rates. .

Larry Summers is worried that this is the dilemma we face: that we are in a world in which r1 is too low…

Thomas Piketty, by contrast, says that he is worried about the world in which r2 is too high.

But it is not r2 but rather r3 that he should be talking about. And r3–the average rate of accumulation–is r2 to which there are a good number of sociopolitical factors plus and minus.

Are Piketty and Summers Reconcilable?

We have a world in which some eminent economists (Larry Summers) say r1 is too low, and other eminent economists (Thomas Piketty) say r2 is too high. Can this compute?


The difference between r1 and r2 is the risk premium. In a well-functioning market economy with well-functioning financial markets, there are powerful reasons to believe that this risk premium should be small: less than 1%-point per year. The fact the risk premium appears to me to be 7%-points per year today is a powerful evidence of the profound dysfunctionality of our financial markets, and of their failure to do their proper catallactic job. But that is a separate and largely independent discussion: that is a dysfunction of our modern market economy which is different from either the dysfunction feared by Summers or the dysfunction freaked by Piketty. For the moment, simply note that it is perfectly possible for all three of these major dysfunctions to occur together.

What Does This Neoclassical Economist Say? Build a Mathematical Model

When a conventional American post-World War II neoclassical economist–somebody, that is, like me–tries to make analytical sense of Piketty’s big book, he says:


No, that’s not it… He says something like:

Piketty talks a lot about eras, and about times when r–his r, r2–r2 > g, and wealth concentration and the wealth-to-annual income ratio is rising, and times when r2 < g, and wealth concentration and the wealth-to-annual-income ratio is falling. But how much? And in what periods, exactly? Let’s see if we can do some finger exercise to figure it out.

We are going to need five eras:

  1. the Agrarian Middle Ages, before 1500;

  2. the Commercial Revolution era, 1500-1800;

  3. the Industrial Revolution/Belle Epoque, 1800-1914;

  4. the “short” Twentieth Century, 1914-1980; and

  5. the Reagan Era: post-1980.

And we are going to need eight quantities, four parameters and four variables:

  1. λ, a parameter: the extent to which as wealth accumulates it competes against itself and allows workers to strike better bargains–the proportional decrease in the average return on wealth r2 that accompanies a proportional increase in the wealth-to-annual-income ratio W/Y.

  2. ρ, a parameter: the standardized bargaining power of wealth in distribution–what the average (risky) return on wealth would be for that society and economy were the wealth-to-annual-income ratio to be equal to 4.

  3. g, a parameter: the trend growth rate of GDP in the economy.

  4. ω, a parameter: the wedge between the average return on wealth and the average rate of accumulation of wealthholders.

  5. r3, a variable: the average rate of accumulation in the society and economy.

  6. r2, a variable: the average (risky) return on wealth in the economy.

  7. W/Y, a variable: the wealth-to-annual-income ratio. And

  8. r2 x W/Y, a variable: the share of income that goes to wealthholders.

There are three questions of great interest for a distributional analysis like that Piketty conducts:

  1. How many years’ worth of annual income wealthholders command: a measure of their social (and economic) power. This is quantity (1).

  2. The share of total income that flow to wealthholders rather than to workers: in any society with inheritance, a measure of the extent to which not merit but position determine relative status and wealth. This is quantity (7).

  3. How broad is the wealthholding class, for if wealth is very broadly distributed high values of (1) and (7) are annoyances rather than worries. But Piketty does not consider this question at all: he assumes that wealth will always be concentrated by the dynamics of luck and family.

Each of our eras will have values for the parameters, and then the dynamic logic of accumulation will drive the values of variables toward a point of equilibrium for those parameter values.

The first thing we know about the equilibrium point is that at it:

r3 = g

If the net rate of accumulation r3 is greater than the economy’s trend GDP growth rate g, the wealth-to-annual-income ratio W/Y is rising, and it will keep rising unless and until changes in other economic variables drive r3 down to g.

r2 = r3 + ω

r3, the net rate of accumulation, related to r2, the rate of return on wealth. But they are not equal. There is a wedge ω between how much income an average wealthholder gets each year from each dollar (franc, pound, mark) of wealth and how much an average wealthholder accumulates each year. War, famine, plague, disease, arbitrary confiscation by the powerful, the requirement of major donations to the church, other liturgical contributions made out of charity or as a way to win a social-status game, conspicuous consumption display, progressive taxes, possible changes in capitalization ratios, and other factors make up ω. It can be very large. What fraction of the return to wealth in Ancient Egypt went to building tombs and pyramids and fighting battles beneath the Hill of Megiddo rather than to accumulation?

There is also another relationship involving r2

r2 = ρ(W/(4Y))^[-λ]

If the economy’s wealth-to-annual-income ratio W/Y were 4, then r2 would be equal to ρ: the standardized measure of the extent to which the society and the economy’s institutions and technologies put the thumb on the scale of rewarding wealthholders in the division of income from production and sales. And as the wealth-to-annual-income ratio W/Y increases, the average return on wealth falls to the extent that capital competes with itself via diminishing returns. The parameter λ captures the extent to which this competition takes place, and thus how rapid is the process by which an increasing wealth-to-annual-income ratio puts downward pressure on the rate of return, and thus on the rate of accumulation, and carries W/Y to an equilibrium point at which g = r3

At this point it is conventional to draw a graph. Conditional on that era’s parameter values for the responsiveness of the rate of return to wealth accumulation λ and the baseline standardized measure of who the society and economy’s institutions put a pro-capital thumb on the scales ρ, there is a downward-sloping relationship between r2 and W/Y:

Piketty Finger Exercises numbers

And the wealth-to-annual-income ratio W/Y will stop rising when the associated value of r2 is pushed down to g, the economy’s trend growth rate, plus ω, the wedge: at that level of W/Y a return on wealth r(2) will be just sufficient as to enable net accumulation at rate g, and so both wealth and income will grow in balance:

Piketty Finger Exercises numbers

In this dynamical system, W/Y will never reach its equilibrium value–but if the eras last for centuries, it will get close, eventually.

And now we can run through our four eras, if we are willing to make assumptions about parameters…

Our First Four Eras

But we do need to make assumptions about parameters. We have four: λ, ρ, g, and ω. We have no good information about λ, so set it equal to 0.5, so that a 1% increase in W/Y calls forth an 0.5% decrease in the rate of profit r2. Thus the rate of profit is neither (a) independent of how much wealth has been accumulated, nor (b) so dependent that increases in accumulation do not actually boost the amount of income going to wealthholders. But there is no justification for choosing this halfway point other than the principle of insufficient reason. For ρ we have nearly a millennium to look at changing W/Y and changing rates of return, so a rate of return of 7%/year seems not unreasonable.

(But if you disagree, play with the model on your own: enter your own preferred values for λ, and ρ, and also for g and ω for the Reagan Era, in the green boxes. And make other changes if you wish–but please play nice, or I will take your toys away…)

The Middle Ages: Before 1500

When we look at northwest Europe back before 1500 we know (a) that the economy’s wealth-to-income ratio was low–there was really not much that was durable and valuable, save for large things made out of stone like cathedrals and castles–(b) that the rate of economic growth was very low: on the order of 0.1%/year; and (c) that life was dangerous: nasty, brutish, and short. g was low because the rate of invention and innovation was low. The rate of invention and innovation was low because not much was known–there were few giants on whose shoulders one could stand–poor societies had low literacy rates, market economies were limited and hobbled, and it was much more attractive to try to make one’s way in the world in the church, the army, or the bureaucracy than to try to invent stuff. A rate of g = 0.1%/year meant that the economy’s wealth-to-income ratio W/Y would grow until r3 was also a mere 0.1%/year. War, famine, plague, disease, arbitrary confiscation by the powerful, the requirement of major donations to the church, and other factors all drove a large wedge between r3 and r2

If we keep our “ignorance” parameters, and if we assign 7.5%/year to ω, we then have:

  • λ = 0.5
  • ρ = 7%/year
  • g = 0.1%/year
  • ω = 7.5%/year
  • r2 = 7.6%/year in equilibrium
  • W/Y = 3.39 in equilibrium
  • r2 x W/Y = 26% in equilibrium

Piketty Finger Exercises numbers

In such a society an enormous proportion of wealth was inherited, one way or another–or simply stolen with or without color of law and authority. After a 30-year generation you would expect to see only 3% more wealth accumulated. The rest? Either inherited; transferred by force, fraud, or contract; or destroyed and either rebuilt or matched by accumulation elsewhere.

This is a picture, but it is a fuzzy picture. Maybe this greatly understates ρ: we have not only market power to boost the profits of capital, but also various forms of extra-economic coercion. Maybe this underestimates the amount of wealth held in forms like castles and cathedrals that are not directly productive in terms of boosting GDP. But it is a place to start, for then starting in 1500 we have the era of the Commercial Revolution in northwest Europe.

The Commercial Revolution, 1500-1800

When we look at northwest Europe, 1500-1800, we see a society and economy growing considerably faster: about 0.5%/year is our consensus value for g. We also see a society with much more law-and-order, and with a restriction of war and arbitrary confiscation to small shares of society relative to the era of private and Hundred Years Wars that had preceded it. If we then raise g to 0.5%/year and assume that the wedge ω drops to 5.0%/year, we then have:

  • λ = 0.5
  • ρ = 7%/year
  • g = 0.5%/year
  • ω = 5%/year
  • r2 = 5.5%/year in equilibrium
  • W/Y = 6.48 in equilibrium
  • r2 x W/Y = 36% in equilibrium

Piketty Finger Exercises numbers

This is the Augustan Age, the Enlightenment Age, the age of Adam Smith–and because “tolerable security of property” allows for accumulation at an extent not possible before 1500, you see lots of authors, not just Smith, writing about how their age is uniquely progressive not or not just because of the progress of science and technology but because of thrift and good order: a civilizing process is underway. But it is also a process that makes society more unequal. I won’t say that there is immiserization at the bottom, but the gains are definitely going to the wealthholders. And that is a process that continues in the era of the Industrial Revolution.

The Industrial Revolution and Belle Epoque, 1800-1914

In this era, the growth rates of western European economies more than double to about 1.2%/year. The great sectoral shift from agriculture to industry which pushes up wages, plus the opening-up of the Atlantic cause a huge wheeling from landed to commercial and industrial property. But at the level of Piketty’s analysis the important thing for the wealth-to-annual-income ratio W/Y is that the increase in the growth rate g is outweighed by a still-further reduction in the wedge ω, with it falling to 3.0%/year or so. Thus we have:

  • λ = 0.5
  • ρ = 7%/year
  • g = 1.2%/year
  • ω = 3%/year
  • r2 = 4.2%/year in equilibrium
  • W/Y = 11.11 in equilibrium
  • r2 x W/Y = 47% in equilibrium

Piketty Finger Exercises numbers

This era lasts for little more than a century. There is, consequently, not time for the wealth-to-annual-income ratio to rise to more than within distant hailing distance of its equilibrium value. And I do not think that the wealthholder share of income in the north Atlantic ever rises to anything close to 47%. For the process is interrupted…

The “Short” Twentieth Century, 1914-1980

World War I. The Bolshevik Revolution. The Great Depression. World War II. The rise of Stalin’s Empire. Plus the coming of social democracy. Yet technological advance, the borrowing of technologies from America, and rapid post-WWII recovery produce a western European average growth rate of 3.0%/year in spite of all the disasters of 1914-1945. But that is not enough to drive a huge reduction in W/Y: you also need wartime destruction, wartime progressive taxation–the conscription to wealth to match the conscription of labor–and postwar turning of the taxes raised to create either a land fit for the heroes who won or a society in which those who lost the war could try to knit things together again. A 3.0%/year value for g, and a return to the pre-1800 5%/year value for the wedge ω give us what we need in order to drive the enormous 1914-1980 decrease in the wealth-to-annual-income ratio that Piketty sees. We have:

  • λ = 0.5
  • ρ = 7%/year
  • g = 3.0%/year
  • ω = 5%/year
  • r2 = 8.0%/year in equilibrium
  • W/Y = 3.06 in equilibrium
  • r2 x W/Y = 25% in equilibrium

Piketty Finger Exercises numbers

This is the middle-class, social democratic society. This is the Kuznetsian normal we thought we had.

The Era of Reagan

And it is Piketty’s main thesis that all of this was upset by the triumph of economic reaction that happened around 1980, with the elections of Reagan and Thatcher, the exhaustion of the social democratic model, the end of the Soviet Empire and thus of any perceived need on the part of the powerful to moderate the expression of their economic power, etc. Piketty[s guesses, if I can put them in quantitative form, are of a slowdown in growth g to 1.5%/year or so and a sharp reduction in progressive taxation that reduce the wedge ω back down to 3.0%/year or so.

This would, in the long run, carry us eventually to:

  • λ = 0.5
  • ρ = 7%/year
  • g = 1.5%/year
  • ω = 3%/year
  • r2 = 4.5%/year in equilibrium
  • W/Y = 9.68 in equilibrium
  • r2 x W/Y = 44% in equilibrium

Piketty Finger Exercises numbers

Of course, we are not there yet. We are at most one generation along the road. The logic of the dynamics says that 30 years is only enough time for us to get about two-fifths of the way from our “short” twentieth century low wealth-to-annual-income ratio to what is the ultimate equilibrium of the current era’s regime. This is Piketty’s forecast: that there is more to come than we have seen in the past thirty-five years.

Is this forecast going to come true? Surely not. For one thing, the political consequences of a narrow close of owners of capital receiving 44% of national income and being able to buy more than nine years’ worth of GDP with their wealth would induce changes of some sort that would alter at least some of the era’s parameters in some way or other.

But if Piketty is right in his forecast of slowing growth, and if he is right in his hope that we will not again see the wealth destruction of the chaos and catastrophes of the twentieth century, then his forecast has the equations on its side. Increase the wedge ω somehow via some positive-sum rather than a negative-sum process, or look forward to a long and large Gilded Age indeed.

Are These Notes Useful?

Are these notes useful? Is this attempt to translate Piketty’s book into what I think of as a Samuelsonian-Solovian dialect enlightening, or is it a creator of confusion? I at least think that I understand Piketty’s big arguments and what they turn upon much better after going through these finger exercises.

Maybe you will too.

You are welcome to try: Feel free to click on, play with, and edit the spreadsheet if you want to. Input values for the four parameters–the dependence of rates of return on accumulation, the baseline economic strength of property, the era’s trend economic growth rate, and the wedge between the rate of return and the rate of accumulation–are in and can be entered into the green boxes…

But do play nice, please…

Piketty Finger Exercises numbers

20140412 Locked Version

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