On r > g and saving rates. Assessing Piketty’s “laws of capitalism” with some simple simulations

In his bestselling book “Capital in the 21st Century”, Thomas Piketty has proposed – alongside his path-breaking empirical work – a small macroeconomic model which has given rise to controversial debates about the evolution of wealth and income inequality.

(eine deutsche Übersetzung des Artikel findet sich hier)

In this Excel file (.xlsx), I propose some simple simulations based on a replication of Piketty’s model which I hope will be helpful even for non-specialists in gaining a feel of what the debates surrounding Piketty’s theory are all about. Specifically, I highlight the importance of differential saving rates (households with high incomes tend to save at higher rates than households with low incomes) for inequality dynamics.

One especially important point of controversy is about the implications of the differential between r, the rate of return on capital, and g, the rate of growth of national income, for the evolution of income and wealth inequality in the model. A common interpretation of Piketty is that whenever r > g (what Piketty calls a “powerful force for divergence”), inequality will rise indefinitely (see the discussion in Larry Summers, “The Inequality Puzzle”.) This interpretation can indeed be based on the following statement made on the first page of the Introduction to “Capital in the 21st Century”:

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 quite likely to do again in the twenty-first, capitalism automatically generates arbitrary and unsustainable inequalities…Thomas Piketty, Capital in the 21st Century

Piketty explains later in the book, that r > g implies ever rising inequality only if the saving rate out of capital income is equal to one, i.e., all capital returns are saved. If this is not so, r needs to be more significantly above g in order for inequality to rise without bound (see also Jakob Kapeller: “The Return of the Rentier” (pdf)). Yet, some economists (e.g., Stefan Homburg, “Critical Remarks on Piketty’s ‘Capital in the 21st Century'” (pdf)) have been quick to dismiss Piketty’s analysis by insisting that it was based on this rather restrictive assumption. And if Piketty is “wrong” on this one, then why bother to deal with the issue of inequality at all? Meanwhile, others have claimed that on the contrary “rising inequality has nothing to do with r > g” (see the discussion in Justin Wolfers, “Inequality and Growth” (pdf)).

In sum, the debate is at times difficult to follow, and probably a bit frustrating especially for non-specialists.

It might therefore be helpful to “play around” with the numerical simulations of Piketty’s model developed in this Excel file (.xlsx). The simulations proposed therein are not necessarily realistic historically, but they may nevertheless be useful for developing some intuition about the relative importance of the key parameters of the model and how they interact in producing different outcomes in terms of income and wealth inequality over different time frames.

An overall conclusion that follows from these simulations, is that the importance of the r-g-differential needs to be put in context. At least as important for the evolution of inequality is that higher income households can be assumed to have higher saving rates (out of lifetime incomes) than lower income households. Although such a notion is both intuitive and can appeal to considerable supportive evidence it is one that many macroeconomic models still ignore (or simply assume away with the representative agent assumption). Homburg’s critique, for example, is based on a model without differential saving rates.

Clearly, if saving rates were independent of relative income, then the wealth-to-income ratio for each household would also be independent of relative income. In the long run, income and wealth inequality would converge towards the wage inequality distribution, and the r-g differential would not matter for income inequality. If high income households save at higher rates than low income households, income will be more unequally distributed than wages. A large r-g differential obviously makes matters worse.

Now, from an empirical perspective, the gap between the saving rates of high income groups and lower income groups has increased in many countries in recent decades, as income inequality has increased. In a recent project funded by the Institute for New Economic Thinking (INET), colleagues and I have tried to analyse why this has happened, building on the Relative Income Hypothesis of Consumption.

For more, take a look at the Excel file (.xlsx).

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