Thomas Piketty’s Questionable Equations

Capital in the Twenty-First Century
by Thomas Piketty
(translated by Arthur Goldhammer)
Belknap Press, 2014, 696 pages, US$39.95


piketty bookNobel Prize winner Paul Krugman in his review in The New York Review of Books describes Piketty’s best-selling book as a “magnificent sweeping meditation on inequality”. Nobel Prize winner Robert Solow writing in the New Republic simply says that “Thomas Piketty is right”. Christopher Croke, writing in The Australian, describes it as a “magnificent 650-page work”. Numbers of other laudatory reviews can be found, particularly from those who look at economics from the left. But from whatever standpoint the book is an impressively wide-ranging commentary on earlier published historical data – painstakingly sourced and constructed by Piketty and his colleagues – on income and wealth going back to the eighteenth century in some cases; across the major countries of Western Europe, and the United States and Canada, and numbers of more ‘farther-afield’ countries, including Australia and New Zealand. However, that is the full extent of its value; iconoclastic though it may be to say so in view of its celebrated status.

Thomas Piketty is a French economist at the Paris School of Economics. From his PhD onwards, obtained at the LSE in the early 1990s, he has specialised on the subject of income and wealth distribution. He spent some time teaching in America at MIT before returning to France. Apparently his work with Emmanuel Saez, a fellow French economist who now teaches at the University of California, Berkeley, provided ammunition for the Occupy Wall Street movement by drawing attention to the wealth and income of the so-called one per cent (‘the top centile’). Piketty is on the left side of the political fence and this is evident throughout his book. Nevertheless, numerous progressive digressions aside, his central thesis is analytical rather than polemical.

Piketty bases his analysis around what, at a fanciful stretch, he calls the two fundamental laws of capitalism. The first law is that the share of national income flowing to capital (α) is equal to the rate of return on capital (r) multiplied by the ratio of the stock of capital to income (β). Thus α = r x β. I have used the same symbols as in the book.

Thankfully, Piketty doesn’t like the intrusion of mathematics into economics so his resort to algebra is of the simplest kind. If, say, the rate of return on capital is 5 per cent and there is six times as much capital as annual income, then capital’s share of national income will be 30 per cent (6 times 5%); with the balance going to labour (in a simplified world of just capital and labour contributing to the generation of income). The equation is in fact an identity. But it is an identity that Piketty works hard in concert with his second law, covered below, to explain economic forces. In fact, he works his so-called laws much too hard to be at all credible, as I will later explain.

His first law takes on explanatory power because his scrutiny of the historical data across time and countries points, he believes, towards the return on capital (r) being relatively constant at around 5 per cent. What this means, if his data and his reading of it are correct, is that the share of national income which accrues to capital largely depends on the size of the capital stock relative to national income. The more the relative size of the capital stock builds the less share of national income will go to labour; and, as capital is tightly held, inequality will rise.

What Piketty finds is that up to the start of the First World War there was growing inequality of wealth. He estimates this reached orders of magnitude of 90 per cent being held by the top ‘decile’ and 50 per cent by the top ‘centile’. National income (wage income plus the return on capital) was also skewed but less so. In France, for example, he estimates that the top decile received 45 to 50 per cent of national income. At the time, he estimates that the capital stock in France, Britain and Germany stood at between 6.5 to 7 times national income. Hence, of course, by ‘the first law’ meaning that a large proportion of national income accrued to the relatively few owners of capital; who could then reinvest the proceeds to further improve their positions.

Piketty asks what would have happened if this process of growing inequality had continued to play itself out through the twentieth century. Could democratic society have survived? Apparently we will never know because a number of factors combined in the first half of the twentieth century to destroy much of the capital stock. According to Piketty, the two world wars, the 1930’s economic depression, low private savings rates, expropriations, and government controls reduced the capital stock to only around 2.5 per cent of national income by the 1950s. Consequently inequality was squeezed.

However, Piketty does not see this as the natural order of things. As a point of departure he leverages off a paper by Simon Kuznets written in the 1950s: “Economic growth and income inequality”. In this paper Kuznets suggests that inequality, like a bell curve, will rise as industrialisation takes off but decline as people more generally participate in industry. Piketty disagrees and suggests that the recent resurgence of inequality since the 1970s – though still leaving wealth inequality much less marked than at the beginning of the twentieth century – is a portent of things to come in the twenty-first century, unless countered by government action. It is necessary to go to his second fundamental law of capitalism.

This law says that in the long run the ratio of the capital stock to national income (β) – on which, as I have explained above, the share of income going to owners of capital depends – is equal to the percentage of income that is saved (s) divided by the rate of growth in national income (g). Thus β = s/g. The rate of saving in the equation is net of capital depreciation.

The sense of the equation is straightforward enough. If savings are a lot more than is required to replace the depreciating capital stock, the capital ratio (β) will tend to rise producing growing inequality. But the central point that Piketty makes is that with any given saving rate, lower income growth also lifts β; and Piketty sees low growth ahead primarily because of very low to zero rates of population growth.

Demography is in fact the force which may undo the democratic social contract by producing insufferable inequality. This is the case because he believes that per-capita growth of income under even the most optimistic scenarios – he assumes 1.5 per cent a year – will be insufficient, without population growth, to create enough income growth to balance the savings rate. The first and second laws will have their way. The ratio of capital to income will grow and produce growing inequality.

“When the rate of return on capital exceeds the rate of growth of output and income as it did in the nineteenth century and seems likely to do again in the twenty-first, capitalism automatically generates arbitrary and unsustainable inequalities that radically undermine the meritocratic values on which democratic societies are based.”

There we have his theory in a nutshell. A trapping is that inheritance will again start having the kind of influence on society and on the economy as it had in the eighteenth and nineteenth centuries, further undermining the association of reward with merit.

There is more to Piketty’s analysis when it comes to the growing wages of CEOs and senior executives. I will cover that later as I will his solutions for countering wealth and income inequality. First to the central question: Is there anything amiss with Piketty’s analysis of the likely growth in the relative size of the capital stock and its implications?

Within the restrictive theoretical bounds of his model represented by his two laws and by his assumptions there is no simple flaw. But, and it is a big but, complex economic forces cannot be modelled with two simple equations and therefore the inevitable implications that he sees on the horizon are far from inevitable. For example, savings can’t go on exceeding the growth in income (as represented in the second law) otherwise we would end up with a superabundance of capital. As capital grows relative to labour you would expect to see the price of capital fall and so reduce the share of income going to those who own it.

Piketty tries to counter this by assuming, on the basis of his interpretation of past data, that the elasticity of substitution of capital for labour is greater than one. Simply put, this means that as the price of capital falls by x per cent; more than an extra x per cent of it will be used in the production process – thus increasing its share of national income. The answer to this is that it is impossible to calculate the elasticity of substitution of such a heterogeneous commodity as aggregate capital. Moreover, common sense points to the elasticity of most components of capital becoming less than one at some early point; otherwise we do end up in an unrealistic economic world of all capital and nothing else. Sometimes common sense must be used to trump untenable and fanciful economic theories.

Piketty also ignores the positive effect of savings on economic growth. Implicitly he has income growing below the saving rate as being a state of equilibrium (i.e., a state which has no tendency to change). But, in fact, saving is likely to be applied to increase growth. You can’t assume that the saving rate and income growth are independent variables. If the rate of saving exceeds income growth for a period it is more than likely that the application of those savings will significantly lift growth. Milton Freidman (Free to Choose) says it well: “The accumulation of physical capital has played an essential role in economic growth”.

A more general case against the two laws is that they operate only in the way described if it is assumed that the shape and extent of technological progress doesn’t shake things up and if the competitive and regulatory landscape stays as is. Once scope is allowed for transforming technological progress and the possibility of substantial product and labour market deregulation (not of course mentioned by Piketty) per-capita income growth might well exceed expectations. Additionally, technological progress, as well as increasing economic growth, is often responsible for the obsolescence of chunks of the existing capital stock; which would, of course, reduce the share of income going to capital. In fact, it seems likely that part of the reason for the relative decline in the capital stock in the twentieth century was because of rapid technological innovations. In short, Piketty’s thesis does not survive real-world scrutiny

Piketty’s two laws do not explain the complex economic forces underlying the progress of capitalism and simply can’t be used to predict the future. It is simplistic to think otherwise. It is also noticeable that Piketty fails to address the upsurge in well-spread prosperity since the Second World War that has accompanied a period of more or less stable inequality and then increasing inequality since the middle 1970s. After all, isn’t it more important that society becomes more prosperous than more equal? Sometimes you might be forgiven for thinking that those on the left value equality over prosperity.

Piketty worries about patrimony becoming increasingly dominant in enshrining wealth among the relatively few. This is understandable if you are mentally occupying the languid worlds created by Jane Austen and Balzac to which he frequently refers. But we are not in that world. To quote Freidman again: “Without the maintenance of inherited capital the gains made by one generation would be dissipated by the next”. It doesn’t matter how people acquire wealth, everyone has the potential of gain, if it is used either directly, or indirectly via investments in stocks and bonds, to increase the productive capacity of the economy. Gina Rinehart is a good case in point but many thousands of others could be identified.

As a completely separate matter Piketty examines the sharp rise in executive income since the 1980s. This has meant that while inequality of wealth is still far less than it was one hundred years’ ago; income inequality has, more or less, returned to its former level. He notes that the rise in executive income has been especially acute in the English speaking countries, with America leading the way; and that this trend will have a compounding effect on wealth inequality. By the way, unsurprisingly, there is nothing on bloated public service levels and salaries.

Valiantly, Piketty makes a show of assessing whether the very large incomes and bonuses paid to CEO’s and senior executives of large corporations can be explained by marginal productivity. In economics, wages (and the rewards to other factors of production) are supposed to be anchored in marginal productivity. Businesses maximise their profits by hiring workers up until their incremental contribution to profits comes close to their wages. Businesses in fact have no way of making such fine calculations. Nevertheless, marginal productivity provides a benchmark and is a valuable pedagogical device. As he rightly should, Piketty rejects any connection between marginal productivity as described in the textbooks and the payment of multimillion dollar incomes and bonuses.

He concludes that the upsurge in executive rewards can be best explained by executives having greater incentives to negotiate harder for more pay if the government extracts far less of it in tax. He notes, for example, in contrast to stability of the top income tax rates in France and Germany (50-60 per cent) that the top rates in the US and Britain fell from 80-90 per cent during the period from 1930 to 1980 down to 30-40 per cent during the period 1980 to 2010; with Reagan bringing the US rate down to a low of 28 per cent in 1986.

Executive rewards these days can be confronting for those of us not earning them. But we need to get above it. The sporting arena can be instructive. Sir Alex Ferguson the former manager of Manchester United football club retired last year. He was the most successful manager in English football history. He was reportedly paid the equivalent of $14 million a year. He was replaced by David Moyse, reportedly paid $9 million, who has just been sacked after ten months of a very unsuccessful tenure. You could say in retrospect that Moyse was overpaid. But was Sir Alex? So far as I know, no-one has ever made that claim. If a second Ferguson were to come along I am fairly confident that he could command an even higher salary than Sir Alex.

Now business is not altogether like sport. But there is a world of difference between, say, Coles falling behind Woolworths or keeping pace or edging ahead. Obviously any number of competitive businesses could be used as examples. Boards and shareholders will be prepared to pay a very large salary to the person who they think will bring success. Failure, as in sport, is extremely painful and costly. It needs to be imprinted in the minds of those on the left that first-rate businesses regularly become second-rate businesses or go out of business and usually because of poor leadership and management.

Global competition is now much more cut-throat than it ever was in the sedate past when technology moved more slowly and when transport costs and tariffs insulated domestic industries. That, I think, is the most important reason why managerial rewards have skyrocketed. Confiscatory top income tax rates might have a dampening effect but it is likely that ways would be found to circumvent more onerous tax rules as they have in the past. It goes without saying that undeserved rewards abound in the corporate world. That’s the imperfect market for you; the least imperfect way to do things. Provided high rewards are not extracted at the point of a gun or by threats and intimidation we should all keep calm and swallow our resentment.

Piketty’s solution to growing inequality, whether of wealth or income is, in a word, ‘taxation’. Certainly high progressive income taxes and inheritance taxes, but most particularly he argues for a global annual wealth tax (which he interchangeably calls a global capital tax). He concedes that this will be near impossible to put in place in the foreseeable future but suggests that it could be approached gradually region by region. By the way, will any of this increased taxation adversely affect output and growth? No committed redistributionist would think so; and therefore it isn’t mentioned.

“The primary purpose of the capital tax is not to finance the social state but to regulate capitalism. The goal first is to reduce inequality of wealth, and the second to impose effective regulation on the financial and banking system in order to avoid crises. To achieve these two ends, the capital tax must first promote democratic and financial transparency: there should be clarity about who owns what assets around the world.”

He believes that the authorities need much more extensive information on holdings and flows of wealth to properly control the financial system. This second goal of the wealth tax sees Piketty straying from his main theme which he does quite often.

There are numbers of progressive digressions in the book: on global warming; on the need to redistribute petroleum revenues in the Middle East; on the need to reduce income inequality among nations; on “new forms of property and democratic control of capitalism”; on “new forms of participation and governance”; on seats for workers on the boards of companies; on the division between capital and labour in an “ideal society”; on “a just social order”; and so on. However, while this stuff firmly nails Piketty’s colours to the mast, with Marx also given twenty mentions, his central thesis remains the analytical one that capitalism has a “logical contradiction very close to what Marx described”. That contradiction is that capitalism consists in building up capital and in so doing creating destructive inequality.

At one level Piketty has constructed an interpretive narrative based on historical data. However that would not have sold so well, I imagine; or caused such a stir. His polemical thoughts on new forms of capitalism and the like are banal to say the least. No, what makes his book ‘interesting and controversial’ are his two interrelated laws purportedly determining the path of capitalism and its almost inevitable creation, left to its own devices, of untenable wealth inequality.

As I explained above, the complexity of an evolving technology-based market economy cannot be captured by two simple equations comprising a few aggregates. Piketty’s thesis is simplistic. I expect free-market academic economists will do a thorough job of dismantling it in the period ahead; if, of course, they think it worthwhile. In the meantime, as examples, two brief commentaries by Hunter Lewis and Peter Klein in the Mises Daily give Piketty’s theory the short shrift it deserves among free market economists.

Why then, you might ask, would numerous renowned economists reviewing this book think it profound? Why would they think that capitalism and its future can be explained by a few aggregate variables and two simple equations? The answer is unfortunately without mystery. They are probably the same economists who live in the equally simplistic world of macroeconomic aggregates created by Keynes. Hayek aptly called this kind of economic thinking the pretence of knowledge when none in fact exists.

We can relax. Capitalism will continue to make the world more prosperous. A freer capitalism would bring more prosperity still. If along the way some do much better than others it is a small price to pay. I will finish with a quote from Hayek, which I am confident those on the left will barely comprehend.

“The market process…corresponds to the definition of a game…It is a contest played according to rules and decided by superior skill, strength or good fortune. It is…both a game of skill as well as a game of chance. Above all, it is a game which serves to elicit from each player the highest worthwhile contribution to the common pool from which each will win an uncertain share.” (“The Atavism of Social Justice”)


Peter Smith wrote “How Money Has Changed, and Why It Matters” in the May issue. His book Bad Economics was published recently by Connor Court.


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