# Is Piketty’s ‘Second Law of Capitalism’ Fundamental?

June 4th, 2014

in Business, Money and Banking, prices - ppi, cpi, etc

*by Per Krusell and Tony Smith, Voxeu.org*

Thomas Piketty’s new book has been widely praised for its empirical contribution, but his prediction of rising inequality rests on economic theory. This column argues that Piketty’s pessimistic forecast is based on an extreme – and unrealistic – assumption about households’ saving behaviour. According to standard theory, the wealth–income ratio would increase only modestly as growth falls, so declining growth would not be a powerful force for generating high inequality.

Follow up:

Over the last several weeks, we have thought quite a bit about the main message in Thomas Piketty’s now world-famous book, *Capital in the Twenty-First Century* (Piketty 2014). We have also discussed it at great length with colleagues. In sum, at least in our departments, there has been a massive collective effort at interpreting both the material presented in the book and the background material on which the book builds. In this column we would like to present one perspective on the book that does not seem to have attracted sufficient attention in the public discussions. We develop these arguments in detail in a separate document (Krusell and Smith 2014).

Piketty takes us on a historical and geographical tour of inequalities in income and wealth. The tour in part feels like a rollercoaster ride. At the very least, it ought to make us all think very hard about the determinants of the levels and trends in inequality, along with possible policy interventions.

- His efforts to collect new historical data and his efforts at measuring wealth – especially among the wealthiest – are laudable.
- The central aspect of the book, however, is actually his prediction for the 21st century – he forecasts a dramatic increase in inequality.

His conclusion is provocative, as is his proposal of a global tax on capital to stem this increase. The problem we have is that we simply do not at all agree with the macroeconomic reasoning that undergirds his forecast.

## What is the cause of our skepticism?

Piketty’s forecast does not rest primarily on an extrapolation of recent trends that he has uncovered in the data. This is what one might have expected, given that so much of the book is devoted to digging up and displaying reliable time series on income and wealth distributions. (Actually, we were surprised to see that there is very little clear evidence of rising wealth inequality in recent years, and even this evidence has recently come under criticism by Chris Giles at the *Financial Times*.) Rather than being based on data, Piketty’s forecast rests primarily on economic theory. We take issue with Piketty’s argument.

The theory in the book is presented in the form of two ‘fundamental laws’, as Piketty dubs them.

- One law is merely a definition – capital’s share of income equals
*rk*/*y*, where*r*is the return on capital,*k*, and*y*is (national) income. - The other law is much more than a definition – it asserts that
*k*/*y*will, in the long run, equal*s*/*g*, where s is the savings rate in the economy and*g*is the sum of the growth rates in population and technology.

Capital’s share of income can therefore be written as *rs*/*g*.

Based on this expression, Piketty then argues as follows. First, the return to capital is insensitive to how much capital is accumulated. As a result, capital’s share of income will be determined largely by *s*/*g*. This, he asserts, will rise rapidly in the future, since most experts expect population growth to fall toward zero and technology growth to decline substantially, too.

Putting this together, regardless of what *s* is, we are poised for a very sharp rise in *s*/*g* (so long as s is positive).

- Piketty considers as plausible that
*g*will be cut at least in half, so his theory implies that capital’s share will at least double. - This doubling would bring us back to levels of inequality similar to those in the 19th century – since capital is today so unevenly distributed.

## Critique

Piketty’s second law is not mathematically incorrect, but it relies on assumptions – as do all economic theories. The central assumption concerns how the economy saves. Piketty assumes that the ‘net’ saving rate is constant and positive, i.e. the economy increases its capital stock from year to year by an amount that is a constant fraction of (net) national income.

This assumption may sound standard but actually it is not – precisely because it is expressed in net terms. In particular:

- With zero growth in population or technology, the assumption that the capital stock is always growing (because net saving is positive) implies that more and more output must be diverted away from consumption towards investment.

Eventually, because capital needs to keep rising, it is necessary to devote 100% of GDP to capital formation!

- Even a fall in growth from 2% to 1% would require that a sharply increasing fraction of GDP be reallocated from consumption to building capital.

We do not think any such reallocations have been observed historically in cases in which growth rates have been persistently low.

Piketty’s theory is closely related to the standard model of economic growth, based on Solow’s pioneering contribution (Solow 1956) – a model that is taught in virtually every intermediate-level undergraduate textbook. This model assumes instead that the gross savings rate, i.e. gross investment (including depreciation) as a fraction of (gross) national income, is constant. Under this assumption, as the growth rate falls to zero, the net savings rate also falls to zero – a sharp contrast with Piketty’s theory. Postwar US data, moreover, is consistent with this theory in that decades with low growth have typically been associated with low (or even negative) net savings rates.

One might argue, of course, that the assumption of a constant gross savings rate is an extreme one that is not grounded in economy theory. There is, of course, an entire field in economics devoted to studying individual consumption behaviour, as well as its aggregate consequences. This literature formulates theories and tests them against data.

The benchmark theory – deriving from contributions by, among others, Friedman (1957), Cass (1965), and Koopmans (1965) – maintains that, at zero growth, capital is maintained at a constant level, i.e. the net saving rate is zero, again in sharp contrast with Piketty’s assumption.

- More generally, the prediction arising out of this literature is that savings rates tend to fall, not rise, as growth falls.
- Neither the textbook Solow model nor a ‘microfounded’ model of growth predicts anything like the drama implied by Piketty’s theory.

In both cases, theory suggests that the wealth–income ratio would increase only modestly as growth falls. Thus, declining overall growth is simply not a powerful force for generating high inequality, and we would not want to make predictions based on it.

## Concluding remarks

We are not sure why Piketty has chosen such an extreme assumption on saving. We have looked in the source materials underlying the book for a test of his assumption, and a comparison with the obvious alternative, and standard, theories, but did not find a clear test or comparison. What is clear, however, is that what lies behind his extreme predictions is his extreme – and, we think, unrealistic – assumption about saving.

Our view, instead, is that wealth dispersion in the Western world – which is very large and most definitely a compelling target of theoretical and empirical study – has primary determinants much different than those emphasised in Capital in the Twenty-First Century. These include, to mention but a few, educational institutions, skill-biased technical change, globalisation, and changes in the structure of capital markets. It is to these forces that those who care about inequality should be devoting their attention, and to which policy reforms ought to be targeted.

## References

- Cass, D (1965), “Optimum Growth in an Aggregative Model of Capital Accumulation”,
*Review of Economic Studies*, 32: 233–240. - Friedman, M (1957),
*A Theory of the Consumption Function*, Princeton, NJ: Princeton University Press. - Koopmans, T C (1965), “On the concept of optimal economic growth”, in
*Study Week on the Econometric Approach to Development Planning*, Amsterdam: North-Holland: 225–287. - Tony Smith (2014), “Is Piketty’s ‘Second Law of Capitalism’ Fundamental?”, mimeo, 28 May.
- Piketty, Thomas (2014),
*Capital in the Twenty-First Century*, Harvard University Press. - Solow, Robert (1956), “A Contribution to the Theory of Economic Growth”,
*Quarterly Journal of Economics*, 70(1): 65–94.