Comparing the Secular Increasing Trend and Effect of the Response to the 2008 Financial Recession on Wealth Inequality in the U.S. with Other Nations Using the Median-based Gini Index

Abstract

Piketty (Capital in the Twenty First Century; Cambridge MA: Belknap Press) and Dorling (Inequality and the 1%; London: Verso) observed that wealth inequality was increasing at a faster rate than income inequality. Wolff (2017) reached the opposite conclusion based on the fact that the Gini index of net worth increased by 9.8% from 1983 to 2016 while the Gini index of income increased by 24.5% over the same period. Because the Gini index does not fully capture changes in a distribution when almost all gains in income or wealth accrue to the upper end, Gastwirth (Statistical Journal of the IAOS; 30:311–320) used a median-based version (G2), which showed that income inequality in the U.S. and Sweden increased at a faster rate than the usual Gini index. Here, analyzing wealth data using G2 shows that wealth inequality increased faster than the Gini index in the U.S. from 1989 to 2019. Similar results are found for Sweden, India and China from 2000 to 2020. A study of the effect of the 2008 financial crisis using the time trend of G2 provides strong evidence that Sweden’s response it decreased wealth inequality, but the reverse was true in the U.S. Canada was the only nation of the six studied where wealth inequality declined over the 2000–2020 period.

Appendix

Appendix A: An Example Illustrating that it takes a Larger Increase in Wealth or Income of the Richest Member of Society to Increase the Gini Index When its Initial Value is High Than When it is Low

Suppose one has data on income or wealth of a population of size n and computes the Gini index (g₁) at time t₁. Using the representation of the numerator of the Gini index as a linear combination of the ordered observations Gastwirth (2017) showed that when the i^tj ranked household of a population of n, receives an increment of size a, which does not change the rank order of households, then the value (g*) of the resulting Gini index is related to the original value (g₁) of the Gini index by

$$g^{*} = g_{1} - \frac{a}{{n\overline{x} + a}}. \left( {g_{1} + \frac{n + 1 - 2i}{{n - 1}}} \right)$$

(1)

Notice that (1) implies that the g* is smallest when the poorest household (i = 1) receives the increment a, and is largest when the richest (i = n) household receives it.

Since we are interested in shifts of the wealth or income distribution in favour of the top, assume that between time t₁ and time t₂ the wealth of the population increases by amount a, which goes entirely to the wealthiest household. Denote the value of the Gini index at t₁ by ${g}_{1}$ and its value at t₂ by ${g}^{*}.$ Setting i = n, (1) becomes

$$g^{*} = g_{1} + \frac{a}{(a + n\overline{x })} \cdot (1 - g_{1} )$$

(2)

There is an interesting interpretation of formula (2), i.e. when the recipient of the highest wealth (or income or whatever is being distributed) receives an additional amount a, while that of the rest of the population remains the same, the Gini index increases by a factor which is the ratio of the increment a to the total wealth of the population at time t₂ times one minus the original value of the Gini index.

One can ask how large the increment a must be in order for the Gini index of a population with total wealth W=$n\overline{x }$ and Gini index ${g}_{1}$ at time t₁ to increase by an amount c, e.g. 0.02, at time t₂? Equating the second term on the right side of Eq. (2) to c yields

$$a = \frac{cW}{{\left( {1 - g_{1} } \right) - c}}$$

(3)

When the total wealth (W) at time t₁ is positive, formula (3) implies that the largest possible increase in value of the original Gini index that can occur when the wealthiest household receives all of the additional wealth a equals 1- ${g}_{1}$. Thus, when the initial Gini index (${g}_{1}$) is 0.8, it requires a larger increment a in order to increase g to 0.85 than it does to increase an initial g of 0.5 to 0.55. Indeed, from A.3, the ratio of the increments that are needed to increase initial Gini indices g₁ and g*, where g₁ < g* by the same amount c equals

$$\frac{{\left( {1 - g_{1} - c} \right)}}{{\left( {1 - g^{*} - c} \right)}}$$

For example, the ratio of the sizes of the increments (a) needed to increase an initial Gini index of 0.8 to 0.85 to that needed to increase an initial index of 5 to 0.55 equals (0.45)/(0.15) = 3. If the initial Gini index equalled 0.9, the corresponding ratio of the increment (a) needed to increase it to 0.95 to that needed to raise an initial index of 0.50 to 0.55 is 9.

Using formula (3) in a similar fashion, one can show that in order for the percentage increase (g*-g₁)/g₁ in the Gini index between the two periods to equal r, the required increment a is given by

$$a = \frac{rW}{{\left( {\frac{{1 - G_{1} }}{{G_{1} }}} \right) - r}}$$

(4)

When the total wealth (W) at time t₁ is positive formula (4) implies that the largest possible percentage increase (r) in an initial Gini index, G₁ is (1-G₁)/G₁. When G₁ = 0.5, r < (1–0.5)/0.5 = 1, but when G₁ = 0.8, r < 0.25. Wolff (2017) noted that between 1983 and 2016 income inequality rose 28%, Since the Gini index of wealth in 1983 was 0.799, r < 0.252; it would be mathematically impossible for the Gini index of wealth to increase by 28%.

From (4) it follows that the increment (a) required to have a 10% (r = 0.1) increase in an initial Gini index of 0.5 equals W/9, but the increment needed for a 10% increase in an initial index of 0.8 is 2 W/3, six times larger than the increment needed to produce the same percentage increase when its initial value = 0.5.

Remark: Regardless of the size of the increment a in formula (2) added to the richest household the term $\frac{a}{ a+ n\overline{x} }$ remains less than one. Thus the value of the Gini index ${g}^{*}$ remains less than or equal to 1.0 when ${g}_{1}$<1.

Appendix B: An Approximate Method for Including the Wealth of the Forbes 400 in the Gini Index Calculated from the Consumer Finance Survey

In order to include the wealth data of the Forbes 400, it is necessary to assume that the wealth of all the individuals in the CFS sample is less than the minimum, C, of the Forbes 400. Let µ₁ and µ₂, denote the mean wealth of the all households in the United States excluding the Forbes 400 and the mean wealth of the Forbes 400, respectively and n₁ = 128.6 million and n₂ is 400, denote the population size of the two groups. Then the Forbes 400 form the fraction γ₂ = 400/(n₁ + 400) of the total number of households, all other households form the fraction γ₁ = 1- γ₂ of all households and the average wealth, µ, of the nation is γ₁ µ₁ + γ₂ µ₂. Similarly, denote the mean difference of the wealth distribution of all households by Δ and the mean differences each of the two groups as Δ₁ and Δ₂, respectively. Using a special case of Yntema’s decomposition of the mean difference, given in Eq. (16) of Gastwirth (1972), the mean difference, Δ, of the entire wealth distribution is given by:

$$\Delta \, = { 2}\gamma_{{1}} \gamma_{{2}} \left( {\mu_{{2}} - \, \mu_{{1}} } \right) \, + \, \gamma_{{1}}^{{2}} \Delta_{{1}} + \, \gamma_{{2}}^{{2}} \Delta_{{2}}$$

(5)

Because Δ is the average absolute difference of the wealth of a random pair of households in the total population formula (16) considers the three types of pairs, one from each group, which occurs with probability 2γ₁ γ₂, both members of the pair are from the total population excluding the Forbes 400, which occurs with probability γ₁² and both are from the Forbes 400, which occurs with probability γ₂². The average absolute difference in wealth of pairs containing one household from each group is µ₂—µ₁. For each of the other types of pairs, the average absolute difference of a random pair equals the mean difference of each group.

The Gini index of the wealth distribution of all households is Δ/2µ. The average wealth of the entire population is the weighted average of the mean wealth of the SCF sample and the mean wealth worth of the Forbes 400. Using these means one obtains the first term in (1). Combined with an estimate of the Gini index of the population covered by the SCF, one can obtain the mean difference of this group and the second term in (1). The mean difference of the Forbes 400 can be calculated from that data and used to obtain the third term in (5). Because γ₂ is very small, the third term makes a negligible contribution to the overall mean difference.

Now consider the data for 2019. The total wealth of the Forbes 400 was 2.96 trillion dollars, implying their average net worth was 7.4 billion dollars. Wolff (2021) reports that the estimated mean wealth of the rest of the population of U.S. households is $728,000 and their Gini index = 0.869. Thus, the estimated mean difference (Δ₁) in wealth for all households excluding the Forbes 400 is $ 1,265,264. In 2019, 128.6 million households were covered in the SCF survey, so the Forbes 400 forms the fraction γ₂ = 3.11*10^–6 of the total number of households. Setting γ₁ = 1- γ₂, the estimated average wealth of all households is $751,015. Notice that including the Forbes 400 increased the average wealth of the nation’s households by about $23,000 even though they form less than one in a hundred thousandth of all households.

Substituting the above estimates into (5) yields the estimated mean difference of the population. The first term is $46,029.5 and the second is $1,265,256. The third term will be quite small because the mean difference of wealth within the Forbes 400 is multiplied by the square of their fraction of the population. For example, if the mean difference equaled 10 billion, the contribution of the term to (5) would be 0.0097, or approximately 0.01. Consequently, its contribution to the Gini index would be less than 10^–8. Thus, the Gini index of the wealth distribution of all households’ equals (46,029.5 + 1,265,256)/ 1,502,030 = 0.873. Thus, including the Forbes 400 increases the Gini index, 0.869, by 0.004.

At first glance, an increase of 0.004 in the value of the Gini index is very small, however, the coefficient 0.00133 of T in the regression of the Gini index of income inequality in Table 7 implies that 0.004 equals the average increase in the Gini index over a three year period.