Nothing Magical about Piketty’s Mathematics

George Cooper worries that Thomas Piketty has advanced some “magical mathematics” in Capital in the Twenty-First Century: mathematics which lead to absurd results. And Cooper argues that when one repairs these absurdities in the most logical way, Piketty’s main policy prescription – a 2% global wealth tax – is seen as a recipe for economically disastrous effects. But the mathematical fix Cooper proposes is not at all plausible given the historical trends Piketty’s research has revealed. More importantly, I think Cooper has simply misconstrued Piketty’s mathematics, and that it doesn’t require any fix at all. When we interpret the mathematical formulae in the way Piketty has indicated, and impose economically natural conditions on their range of application, Cooper’s problem disappears, and Piketty’s historical analysis and projections emerge unscathed.

The problem Cooper believes he has detected emerges from what Piketty calls the Second Fundamental Law of Capitalism. The law relates an economy’s long-term capital-to-income ratio β to its savings rate and growth rate. For convenience, Piketty states this law in the form “β = s/g”.  Since β is the ratio of a nation’s capital K to its annual national income Y, it might seem that the law can be restated in the form of the identity:

K/Y = s/g

And if we do interpret it as an identity – or at least an approximate identity – the Second Fundamental Law would lead to some rather absurd results for hypothetical low-growth economies. For example, using one of Cooper’s examples of an economy with a 10% savings rate and 0.25% growth rate, we get an s/g of 40. If K/Y is approximately equal to s/g, and if the rate of return on capital is between 4% and 5%, then the capital share of income rK/Y falls approximately between 160% and 200%. Obviously income from capital cannot exceed total income, so something has gone wrong here.

What Cooper thinks has gone wrong is Piketty’s claim that the rate of return on capital r can persist at its historically observed rates of between 4% and 5% into the future, even if we enter an economic environment of very low growth. He thinks Piketty’s equations “simply cannot hold true in the low growth environment which he is trying to analyse,” and that these equations therefore need to be fixed by introducing different assumptions about the relationship between r and g. He offers such a fix:

The question is how to fix them. The most logical approach is to accept that the yields on assets fluctuate to reflect the growth rate of the economy. If growth is cut in half then asset prices will double but their yields will also be cut in half, a condition met when r = g.

But this argument has gone quite off the rails due to a mistaken interpretation of the Second Fundamental Law, because as Piketty makes clear in his discussion of the law, the Second Fundamental Law of Capitalism is neither an identity nor or an approximate identity, but a long-term asymptotic law. We might therefore choose to write it less misleadingly this way:

β → s/g

The import of this law is that, for any fixed national savings rate s and fixed growth rate g sustained over some period of years, β will be moving in the direction of s/g during those years. If β is less than s/g, then the rate of wealth growth will exceed the rate of income growth, and β will increase. If β is greater than s/g, then the rate of wealth growth will be less than the rate of income growth, and β will decrease. Piketty employs this long-term law to explain why some countries which save a lot and grow slowly accumulate large stocks of capital relative to income, and how “small variations in the rate of growth can have very large effects on the capital/income ratio over the long term.” The law thus helps to explain “capital’s comeback” in recent decades.

Cooper and others seem to have come away from a cursory reading of Capital in the Twenty-First Century with the impression that Piketty is claiming that β is always approximately equal to s/g. But that is not true. As Piketty points out, if we start with capital-to-income ratios sufficiently far from s/g, then even if the prevailing rates s and g are sustained indefinitely, it might be a very long time indeed before β gets near s/g. For example, if a national economy possesses a capital-to-income ratio of 6, a savings rate of 10% and a very strong growth rate of 5%, then s/g will be 2. But even if it maintains those savings and growth rates indefinitely, it would still take 58 years for β to fall below 2.25, and 92 years for it to fall below 2.05. The lower the growth rate, the slower the speed of convergence. If we start with the same capital-to-income ratio and savings rate just posited, but assume a 1% growth rate, then s/g is 10. But it will take 140 years for β even to exceed 9, and 281 years to exceed 9.75.

Now, obviously, indefinitely fixed savings rates and growth rates are economically unrealistic for societies with very low growth rates or very high savings rates. For the sake of thought experiment, imagine an ultra-low growth economy: for example, a society that has settled into a growth rate of 1/100th of a percent. This is effectively a steady state economy. The Second Fundamental Law seems to say that the capital-to-income ratio would increase continually and approach 1000. But, that is absurd. No real economy would behave in such a way. A society committed to a roughly steady state would accumulate capital sufficient to produce its more-or-less fixed annual output, with some additional capital hoards accumulated as a hedge against unforeseen emergencies, and save only what it needed to save each year to replenish depleted or degraded capital stock. It would likely settle on some target capital-to income ratio βT and reduce its savings rate gradually to bring it down close to g x βT. But these reflections aren’t really in tension with the Second Fundamental Law, since that law only describes the direction of long term capital accumulation trends in the real world in the presence of relatively consistent savings and growth rates.

When combining these theoretical conclusions relating to savings, growth and the capital-to-income ratio with further posits about the rate of return to capital in the description of a hypothetical economy, we need to impose some economically realistic further conditions, even in the case of extremely idealized models, if the posited rates are to describe a possible real-world economy. For example, since the total annual return to capital is rK, and that return can never exceed Y in any economy, then r must always be less than or equal to 1/β.  Employing the Second Fundamental Law, another way of putting this is that it is that since 1/β continually approaches g/s, it is not possible to have even a fantasy economy with  indefinitely fixed g, fixed s and fixed r, where r exceeds g/s.  If these are the initial values, something would have to give eventually.

Nevertheless, for real world economies of the kind for which we have historical experience, rates of return on capital within the historically observed 4% to 5% range can be sustained indefinitely. Indeed, if an economy stabilized with a β of 6, and with relatively stable rates of growth and savings, it could sustain any return on capital less than 16.7%. Even an economy with a β of 10, a value far above the normal range for actually existing economies of the past few centuries, would permit a sustainable return to capital of 10%. Since Piketty observes a consistent 4% to 5% return to capital over this time, there is no tension whatsoever between his observed results and the mathematical model. Nor is there any mathematical reason to doubt that r can, and will, exceed g by a substantial amount in the future, leading to strong forces of income divergence and increasing inequality.

But now, let’s descend out of these mathematical clouds, and our discussions about what is merely mathematically possible for idealized economies, and focus our attention on what is true of the economies we actually live in, and on how they are likely to evolve in the near future. It is extremely important to remember that Piketty’s model is not a deterministic system from which he attempts to predict all future economic history, but rather a system of interacting mathematical regularities and patterns, themselves directly measurable from the statistical analysis of historical data, intended to give a good match to empirically observed results, and from which we can then make some predictions about the future by extrapolating the most robust trends and incorporating what we know of present economic conditions. In much of the recent critical commentary on Piketty’s book, however, we find Piketty’s fundamentally empiricist approach to economics crashing again and again into the deeply engrained scholastic rationalism of contemporary economics, or into the so-called economic “positivism” of theorists who build models that can approximate some empirically observed macroeconomic facts by continually re-calibrating the values of theoretical parameters that are obviously and deeply unrealistic.

Piketty’s Capital in the 21st Century is not primarily a work of theory and mathematical modeling, although it does contain important elements of those things, but rather a data-driven study of over two centuries of historical economic reality. Returns on capital consistently between 4% and 5% are an observed economic fact; the presence of such high returns even in conditions of relatively low growth, with r exceeding g by substantial amounts for long periods of time, is an observed fact; growing inequality during such periods is an observed fact; the reduction in inequality during long stretches of the 20th century coincident with the capital destruction caused by the world wars and with the redistributive – sometimes radically redistributive – politics of that era is a fact; and the increased role of capital and capital income, along with growing inequality, in the last decades of the 20th century and the first part of the 21st century, are observable facts.

People who wish to criticize Piketty’s projections and policy recommendations need to develop economic scenarios which are both mathematically plausible and empirically consonant with historical patterns and contemporary conditions. There is of course no logical guarantee that the future will be like the past. But if critics wish to avoid Piketty’s conditional projections, and argue that the future is suddenly going to veer into a very different direction, then the burden is on the critics to provide a plausible account of which actual economic and political conditions might bring about these changes. It is not enough to describe scenarios that are merely mathematically possible, but have no high level of probability given what we observe of prevailing conditions.

It is also important to note that Piketty’s inequality r > g is not responsible by itself for the strong forces of contemporary income divergence and growing economic inequality. Whether r > g is producing inequality or not depends on dynamics within the society. In a society, for example, in which every adult had an equal ownership stake in the society’s capital stock and capital income, then only differences in returns on labor would contribute to growing or shrinking economic inequality. What Piketty points out is that, given the unequal distributions of capital that actually obtain, r > g is a force for greater wealth and income divergence. And the greater r exceeds g, the stronger these forces will be. There are other factors driving inequality as well, including the rise of “super-managers” and their super-salaries, a phenomenon that Piketty argues is taking place mainly in the English-speaking world. Piketty’s full analysis of the structure of contemporary inequality, and its likely future evolution if the forces driving it are not altered by policy, is not derived mechanically from a couple of simple laws, but is developed over six chapters of the book.

In considering the possibility of replacing Piketty’s inequality r > g with the equality r = g, Cooper says, “I expect the r = g assumption will make more intuitive sense to investors who have seen the real yields on, for example, inflation protected bonds collapse as growth has fallen.” But, Capital in the Twenty-First Century is primarily the fruit of 15 years of exhaustive historical study. If some investors possess intuitions about the overall rate of return on capital based on the behavior of some assets in their own personal investment portfolios, then at this stage I think we have to say that Piketty has shown those intuitions are seriously flawed.

16 thoughts on “Nothing Magical about Piketty’s Mathematics

  1. Pingback: Summers’s Review of Piketty: Underestimating the Argument for the Forces Driving Inequality | Rugged Egalitarianism

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  3. Reblogged this on Political Economics Review Blog and commented:
    Critique of Piketty (pro-right-wing thinkers I guess!). Heared a lot of writing about th issues rised by this writing piece….but I think it’s worthy to post (you can Judge yourself wether the arguments put forward are consistent, objective, and right)!

    Like

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