Appendix A: Measuring richness
1. Methods
Recently richness indexes have been introduced to assess a typical problem of “headcount” indexes of the rich, i.e. their insensitiveness to changes in the income/wealth of those already classified as rich (Peichl et al. 2010, 598). The newrichness indexes are analogous to some well-known measures of poverty - they are, in fact, the adaptation to richness of pre-existing indicators of poverty (Medeiros 2006; Peichl et al. 2010; Bose et al. 2014). As, generally speaking, they take into account the gap existing between the rich and the others, they do not only measure the number of rich people (as with headcount indexes) but also the intensity of richness.
It is beyond the scope of this Appendix to discuss in detail the properties that a desirable richness index should have - a topic which is analyzed in the specialized literature (see for example Peichl et al. 2010; Bose et al. 2014). It is, however, worth underlining the fact that proper richness indexes should satisfy the so-called "Focus axiom", according to which the index should be independent from the incomes (or the wealth) of the non-rich. This is a particularly useful feature for a study of preindustrial times, when the conditions of the poor are usually impossible to reconstruct other than purely hypothetically. This does not mean that the absence of the poor has no impact on the values acquired by a richness index – but that such impact is mediated through the fixation of the value of , the richness line. Were it possible to establishat an absolute level independent from the distribution – say, 50 times the subsistence income -, then the value of the richness index would not be altered in any way by the inclusion or absence of the poor in the distribution. But as in most cases, for practical reasons, is calculated starting with the information available about the distribution - say, as a given multiple of the median or mean income or wealth -, the presence or absence of part or all of the poor will have some impact on the value of the indexes.
From the many richness indexes described in the literature, I decided to consider only those that can be standardized[1] to vary between the values zero and one. In fact, given the particularly long time periods considered and the variability in time and in space of the units of measurement with which wealth is expressed in our sources, standardized richness indexes help to provide a more coherent and reliable overall reconstruction.
Following Peichl et al. (2010, 603), from the family of richness indexes I selected one that is analogous to the widely-used poverty index of Chakravarty (1983):
When , resembles , the headcount index. Regarding the meaning of , it should be noted that its value determines the shape of the individual affluence function, f. When is small (say, below 0.4), the value of the index increases slowly as the income or wealth of a rich individual increases. When is large (say, 3 or more), an increase in the income/wealth of a rich individual quickly brings his or her contribution to the index close to the theoretical maximum of (see Peichl et al., 2010, 603 fig.1 for a graphical representation).
2. Robustness checks
The analyses presented here integrate those discussed in the article main text (section 6). In particular, they provide a robustness check of the results to different values of β. In fact, the β parameter determines the actual value assumed by the richness indexes (see discussion above). In particular, the higher the β, the larger the increase in the index as the wealth/income of a specific individual/household increases. One could wonder, then, if the trend described in Figure 5 of the main text stands only for specific values of β. To this end, I conducted extensive robustness checks which confirm that the trend is not “artificial” but truly reflects social and economic change. This can be seen in Figure A1, where for the Sabaudian State (cities only – the longer time series used here) different values of β are used (from 0.1 to 10), as well as in Table A1, in which for all the communities or areas included in earlier tables, richness indexes for the most commonly-used values of β are provided (0.3, 1 and 3).
Figure A1. Robustness of the general trends to changes in the value of β: the case of the Sabaudian State, cities only (. Richness line =1000% of median)
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Table A1. Long-term developments in “richness” in Italy and Europe, 1400-1800
Sabaudian State / Sabaudian State (citiesonly) / Florentine State / Kingdom of Naples (Apulia) / Southern Low Countries (cities only)= 0.3 / = 1 / =3 / = 0.3 / = 1 / =3 / = 0.3 / = 1 / =3 / = 0.3 / = 1 / =3 / = 0.3 / = 1 / =3
1300 / 0.012 / 0.030 / 0.046
1350 / 0.008 / 0.020 / 0.036
1400 / 0.005 / 0.013 / 0.025 / 0.007 / 0.016 / 0.028
1450 / 0.003 / 0.008 / 0.014 / 0.005 / 0.013 / 0.021
1500 / 0.004 / 0.010 / 0.019 / 0.006 / 0.015 / 0.027 / 0.007 / 0.018 / 0.032
1550 / 0.003 / 0.009 / 0.017 / 0.007 / 0.017 / 0.031 / 0.008 / 0.020 / 0.035 / 0.004 / 0.011 / 0.022
1600 / 0.006 / 0.017 / 0.030 / 0.008 / 0.020 / 0.037 / 0.011 / 0.026 / 0.040 / 0.013 / 0.032 / 0.054 / 0.005 / 0.013 / 0.025
1650 / 0.009 / 0.022 / 0.037 / 0.015 / 0.036 / 0.058 / 0.014 / 0.032 / 0.054 / 0.011 / 0.025 / 0.042 / 0.003 / 0.010 / 0.019
1700 / 0.011 / 0.027 / 0.045 / 0.020 / 0.048 / 0.077 / 0.021 / 0.049 / 0.075 / 0.014 / 0.032 / 0.050 / 0.004 / 0.011 / 0.023
1750 / 0.018 / 0.041 / 0.063 / 0.020 / 0.049 / 0.079 / 0.033 / 0.072 / 0.107 / 0.017 / 0.037 / 0.058 / 0.006 / 0.017 / 0.033
1800 / 0.025 / 0.056 / 0.084 / 0.025 / 0.059 / 0.089 / 0.008 / 0.021 / 0.040
Bergamo / Padua (city) / Padua (contado) / Padua (city and contado) / Cervera / Reus
= 0.3 / = 1 / =3 / = 0.3 / = 1 / =3 / = 0.3 / = 1 / =3 / = 0.3 / = 1 / =3 / = 0.3 / = 1 / =3 / = 0.3 / = 1 / =3
1400 / 0.012 / 0.030 / 0.052 / 0.000 / 0.001 / 0.004
1450 / 0.011 / 0.028 / 0.049 / 0.001 / 0.001 / 0.003
1500 / 0.012 / 0.030 / 0.049 / 0.004 / 0.009 / 0.018 / 0.002 / 0.004 / 0.007
1550 / 0.019 / 0.045 / 0.075 / 0.019 / 0.045 / 0.073 / 0.007 / 0.016 / 0.028 / 0.026 / 0.060 / 0.092 / 0.006 / 0.015 / 0.027 / 0.003 / 0.007 / 0.013
1600 / 0.017 / 0.044 / 0.074 / 0.041 / 0.091 / 0.133 / 0.013 / 0.031 / 0.050 / 0.032 / 0.070 / 0.104 / 0.002 / 0.004 / 0.007
1650 / 0.014 / 0.035 / 0.060 / 0.036 / 0.082 / 0.122 / 0.013 / 0.032 / 0.051 / 0.037 / 0.081 / 0.117 / 0.002 / 0.006 / 0.014 / 0.002 / 0.005 / 0.008
1700 / 0.026 / 0.062 / 0.098 / 0.032 / 0.074 / 0.111 / 0.017 / 0.041 / 0.065 / 0.041 / 0.089 / 0.128 / 0.005 / 0.014 / 0.026 / 0.003 / 0.007 / 0.012
1750 / 0.003 / 0.009 / 0.016
1800 / 0.007 / 0.018 / 0.030
Notes: Measures organized around reference years when needed. For actual years, see table 1 in the main text. The measures refer to wealth distributions (excluding those with no property) for all areas except for the southern Low Countries where it refers to income distributions. The richness line is set at 1000% of median (500% for Southern Low Countries)
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References to the Online Appendix
Bose A, Chakravarty SR, D’Ambrosio C (2014) Richness orderings. The Journal of Economic Inequality 12(1): 5-22
Chakravarty SR (1983) A New Index of Poverty. Mathematical Social Sciences 6: 307–13.
Medeiros M (2006) The Rich and the Poor: The Construction of an Affluence Line from the Poverty Line. Social Indicators Research 78: 1-18.
Peichl A, Schaefer T, Scheicher T (2010) Measuring richness and poverty: a micro data application to Europe and Germany. Review of Income and Wealth 56(3): 597-619
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[1]Not all potentially useful richness indexes can be standardized, essentially due to the fact that while for poverty measurements, there is a clear boundary to poverty (zero income or wealth), this is not the case with richness whose maximum value is not clearly determined. Only a subset of richness indexes (those characterized by a concave individual affluence function, f) can be standardized (Peichl et al. 2010).