income shares based on tabulations of income tax records for a large number of countries, based on the work of Anthony Atkinson, Thomas Piketty, and their collaborators.<sup>15</sup> We retrieve time series data on mean income and the income share of the bottom 90 or 99 percent of income earners (depending on data availability), for a set of 19 countries with long annual time series data on these aggregate moments in the post World War II period.<sup>16</sup> We convert these two summary statistics into the mean and variance of log income using our maintained assumption of lognormality.<sup>17</sup> The WID predominantly contains data from advanced economies, and our set of 19 countries from this source includes only three developing countries: China, India, and Mauritius.

Our task of estimating � based on the evolution over time of these aggregate moments is complicated by the fact that the time period covered for many of these countries is long, and likely spans some structural breaks. Most of the countries in our sample have data beginning shortly after World War II, and visual inspection of trends in the mean and variance of log income suggest a trend break in these series around the 1970s for many countries (see Figure 2). Three countries in our sample (United States, France, and Germany) also have fairly long time series data prior to World War II, where again it seems plausible that income dynamics may have been different relative to the post-war period. We therefore divide the data into sub-periods, and allow for different time trends by sub-period. For the three countries with pre-World War II data, we consider the available data up to 1939 as one distinct period. For the postWorld War II period, we allow the data to select a single structural break in the time trends. Figure 2 illustrates this process of identifying trend breaks for different time periods for the three countries in our sample with pre-World War II data. The left and right panels of the figure report the time series of the mean and standard deviation of log income, with the trend lines for different sub-periods superimposed.

We impose the restriction that � is the same over the entire time-series available for each country, but we allow the intercept and time trend in Equations (8) and (9) to differ across the sub-periods for each

> 15 See for example Atkinson and Piketty (2007, 2010), Atkinson et al. (2011), Banerjee and Piketty (2005), and Roine and Waldenstrom (2008).

> 16 Our default is to use the bottom 90 percent share when available.  We use data on bottom 99 percent for 5 countries (United Kingdom, India, Japan, Mauritius, and Singapore), for which data on bottom 90 percent is either limited or not available in the WID database.

> 17 While the top incomes data in the WID is based on tabulated tax records, mean income is an estimate of income of all individuals including those who do not file tax returns, often based on national accounts measures of household income (see e.g. Atkinson et al. (2011) for details).  The WID report data for income shares higher than the top 10% that we use here, and the very top income shares are based on fitting a Pareto distribution to the highest observed income groups.  We use the top 10% share since it is least likely to reflect the Pareto imputation of the top tail of the income distribution, and therefore is more likely to be consistent with our lognormality assumption.

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