Piketty's book * Capital in the twenty-first century* is, in the author's own words, “… a book about the history of the distribution of income and wealth.” Among other interesting and important facts, the book quantifies the evolution of wealth inequality and wealth concentration over time and across a number of countries. Wealth is highly concentrated and its distribution is skewed with a long right tail (see also Atkinson 1983, Wolff 1995, 1998, and Rodriguez 1998) – a small number of very rich individuals hold a large share of total wealth in the economy. The book documents that the share of aggregate wealth in the hands of the richest individuals displays a U shape over time, trending downward for most of the 20^{th} century and then increasing from the 1980s onward (see Figure 1). In other words, wealth has become more concentrated over the past 35 years.

**Figure 1**. Wealth inequality: Europe and the US, 1810-2010

Though the book discusses a number of mechanisms affecting wealth inequality – the role of tax progressivity, top income shares, and heterogeneity in saving rates and inheritances – it singles out a ‘fundamental force for divergence’ in the size of the difference between the post-tax rate of return on capital and the rate of output growth (see Figure 2). According to this mechanism, a higher post-tax rate of return increases the rate at which past accumulated wealth compounds, thus magnifying wealth inequality. Conversely, a higher rate of output growth reduces wealth concentration by increasing labour earnings and, therefore, saving by individuals whose main source of income is labour earnings. In Piketty's view, the effect of these two forces is big, and changes in the rate of capital taxation and output growth can explain the dramatic evolution of wealth concentration over the past century. It should be noted that according to Piketty it is the *difference* between the net rate of return on wealth and the output growth rate that affects wealth inequality. Furthermore, he does not distinguish between changes in the rate of output growth due to changes in total factor productivity (TFP) versus those in the population growth rate.

**Figure 2**. After tax rate of return vs. growth rate at the world level, from Antiquity until 2100

In a recent paper on Piketty’s book and macroeconomic models of wealth inequality, we provide two contributions. First, we take stock of the existing literature on models of wealth inequality through the lenses of the facts and ideas in Piketty's book and highlight both what we have learned so far and what we still need to learn in order to reach more definitive conclusions on the mechanisms shaping wealth concentration. Second, we explore the quantitative importance of the mechanism proposed by Piketty by evaluating the effect of changes in the rate of return on capital and the rate of output growth on wealth inequality in a quantitative model capable of generating realistic wealth inequality.

In the first part – that takes stock of the existing models of wealth inequality – we initially discuss the (mostly analytical) literature aiming to account for the observation that the right tail of the wealth distribution is well approximated by a Pareto distribution (See, e.g., Benhabib *et al *2011, and Aoki and Nirei 2015, for two recent contributions). This strand of the literature provides the main theoretical underpinning for the mechanism, emphasized in Piketty's book, according to which wealth concentration increases with the difference between the average net rate of return on wealth *r* and the trend rate of growth of aggregate output *g*. Multiplicative idiosyncratic random shocks to the rate of return on wealth are the main mechanism that generates wealth concentration in this class of models. While Piketty sees the rate of output growth as unambiguously reducing wealth concentration, according to some of these models output growth due to TFP increases can either reduce or increase wealth concentration depending on the environment (e.g., Aoki and Nirei 2015). For tractability, this literature abstracts from key aspects of reality, such as the determinants of the heterogeneity in rates of return (for instance due to entrepreneurship and portfolio choice), the life cycle, and the observation that bequests are luxury goods (consistent with the evidence that saving rates are strongly increasing in wealth, as documented by Dynan *et al *1993, and, more recently, Saez and Zucman 2014). In addition, Gabaix *et al* (2015) show that these models imply a transitional dynamics of wealth concentration that is much too slow to account for its empirical evolution over the last 35 years.

Endogenous heterogeneity in saving rates in response to earnings and expenditure shocks (including, possibly, medical and nursing home expenditures during retirement), and in some cases endogenous heterogeneity in rates of returns, are instead at the centre of the quantitative models that we discuss next. The comparative advantage of this literature is its emphasis on understanding the forces that shape differences in saving behaviour and rates of return and on quantifying the importance of such heterogeneity in accounting for wealth inequality in rich quantitative models. Previous work has convincingly emphasized that entrepreneurial activity (Cagetti and de Nardi 2006), a luxury-good bequest motive (De Nardi 2004), heterogeneity in patience across families (Krusell and Smith 1998, Hendricks 2007), and high earnings risk for top earners (Castaneda *et al* 2003) help explain the high degree of wealth concentration (De Nardi *et al* 2015 investigate the latter point using tax data on earnings). Among these, entrepreneurial activity and luxury bequests appear to be the most promising. However, it is not clear to what extent each of these forces interacts with the others and jointly contributes to wealth inequality because, at least so far (with the exception of De Nardi and Yang 2015), most of these forces have been studied in isolation. There is also work to do in determining to what extent these quantitative frameworks can match the observed large differences in wealth inequality, both across countries and over time. Promising work by Kaymark and Poschke (2015) shows that these models, appropriately matched to data, succeed in explaining the evolution of wealth inequality and top wealth shares in the US over the last 50 years. More specifically, they show that increasing wage dispersion during this period is a major driver of trends in inequality and that changes in taxes and transfers to seniors account for nearly half the observed increase in the wealth concentration.

In our second part, we assess both the strength of the *(r-g)* force and the extent to which, as conjectured by Piketty, it is the *difference *between the rate of return on capital and output growth that drives wealth concentration. We carry out our analysis within a rich quantitative model that can account for the observed inequality in both wealth and earnings. We show that changes in the rate of return on capital and TFP have only small effects on wealth inequality, while changes in output growth due to population growth have very large effects. The key intuition is that the rate of return and the TFP growth rate affect all households in a similar way. In contrast, a change in the population growth rate affects different categories of individuals through two main channels. First, changes in the population growth rate affect the number of people among whom a bequest is divided and therefore the average bequest size. If, for instance, the population growth rate decreases, inherited estates tend to be larger and since richer people leave larger bequests, a reduction in population growth generates more wealth inequality. Second, a fall in the population growth rate changes the demographic structure and in particular it increases the fraction of older people, who tend to be richer. Thus, the rate of return on capital and of output growth are not perfect substitutes in their effect on wealth concentration when output growth is due to population growth. In fact, an increase in the rate of return on capital raises wealth concentration substantially less than a fall in the rate of population and output growth by the same amount.

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