Ifo Viewpoint No. 158: Piketty’s World Formula

Hans-Werner Sinn
Munich, 14 May 2014

Thomas Piketty’s book on inequality hit a sensitive spot with the Americans. It channelled the mounting dissatisfaction in a country that lacks a robust welfare system and a more progressive tax system. 

His book has a whiff of Karl Marx, with Piketty’s style of writing and his resorting to a theory similar to Marx’s when he decries the increasing inequality that results from an ever higher ratio of wealth to national income. Marx had already prognosticated the increase in this ratio with his law on the growing organic composition of capital.

Piketty attributes the growth in inequality to the r > g formula, one of the few formulas that have made it to the international daily papers and which by now has attained a hallowed status among journalists akin to Einstein’s e = mc2. The formula states that the interest in the form of the average return to capital (r) is persistently higher than the growth rate of the economy (g). The consequence, according to Piketty, is that the wealth of an economy accumulates faster than the growth in economic output. Piketty’s world formula is a hot topic of discussion everywhere.

In fact, the formula has been known quite long now; it denotes a fundamental assumption of economic growth theory. Indeed, over the long run the rate of return to wealth usually lies above the growth rate of the economy, as Piketty asserts, for if not, land prices would be infinite, there would be excessive consumption, and growth would eventually end. But that does not mean that wealth grows faster than economic output. Such a conclusion would only be warranted if the savings of an economy could be set equal to the interest income, so that the rate of economic growth is the same as that of wealth. But that is not the case. Rather, savings are consistently smaller than the sum of all capital income. The wealthy consume substantial parts of their income, and the savings from labour income usually is small. Thus, the growth rate of wealth lies significantly below the interest rate, and the fact that the interest rate exceeds the rate of economic growth does in no way imply that wealth grows faster than the economy.

As a matter of fact, it is a central finding of economic growth theory that the interest rate of an economy, dependent on the savings rate, settles over the long term at a level in which the growth of capital equals the economic growth rate. The consequence is the long-term persistence of the ratio of wealth to economic output. The long-term constancy of this ratio is a basic ingredient of all growth theories.

Behind the long-term persistence of this ratio stands a simple mathematical law. If an economy saves a given portion of its income, the wealth represented by the accumulation of these savings will increase in the long run at the same rate at which national income grows. The ratio of wealth to income cannot thus increase permanently.

The law is based on the fact that every increasing quantity can grow over the long run only at the rate at which its accretion grows. An example is the heaping of earth into a mound. Assume that in every period, a further spade of earth is added, and that the size of the spade itself grows at a given rate from one period to the next. The growth rate of the amount of earth in the mound converges to the growth rate of the spade size. If we substitute the current savings of an economy for the amount of earth in the spade and wealth for the size of the mound, we obtain the long-run constancy of the ratio of wealth to income when a fixed share of income is saved.

It must be stressed that this law applies over the long run, over several decades. Wealth can well grow faster than the economy at given times. Piketty could then have a point.

But even when such is the case there is hardly any reason for apprehension, since when it comes to distributional issues the important element is less the ratio of wealth to national income than the ratio of capital income to wage income, that is, the proportion of capital and wages in national income. This ratio, as had first been observed by the left-leaning economist Joan Robinson in her famous book An Essay on Marxian Economics in 1942, has remained fairly stable over time and follows no discernible trend.

Much more important than Piketty’s theory of everything is the question of how many people share in the wage and capital income. If the number of wage earners increases faster than the number of wealth owners, a less desirable distributional pattern could emerge despite the constancy of the ratio of capital to wage income. This could be the case in the US, with its large number of immigrants, and could be the reason for the current dissatisfaction among the populace. But there is no evidence to support this as a general law.

And if the risk should indeed exist that the number of rich people compared to that of poor people grows too slowly, the best medicine is to improve the upward-mobility chances. The more rags-toriches occurrences, the smaller the distributional problem.

It also helps if the rich have more children than the poor, since their wealth will eventually become spread among their heirs, solving the distributional problem at a stroke. A family income splitting system such as France’s is one of the policy measures that a society might consider if it fears an undesirable concentration of wealth.

Irrespective of the above, a progressive taxation system is needed to check the growth in net income among the upper income echelons, since even if there is no fundamental trend towards greater inequality owing to the theory formulated by Piketty, inequality within the wealthy group can increase because individual dynasties accumulate ever more wealth. Whether there is a particular need for action in this regard in Europe is open to debate, since progressive taxation has attained here a significant degree already.

The conviction remains that Piketty, like Marx, caters to a longing simmering among the people, but that he tries to underpin his policy proposals with a theory that does not substantiate what he asserts.

Hans-Werner Sinn
Professor of Economics and Public Finance
President of the Ifo Institute

Published in German under the title “Thomas Pikettys Weltformel”, Frankfurter Allgemeine Sonntagszeitung, No. 19, 11 May 2014.