By looking at the distribution of wealth among the agents we were able to conclude that various forms of taxation and distribution can lead to greater or lesser wealth equality. However, it is useful to introduce an explicit measure of wealth inequality so that we can reach more quantitative conclusions.

A common measure of wealth inequality is the Gini coefficient (or Gini index), G.

A small value of G indicates a more equal wealth distribution with G = 0 corresponding to complete equality. In contrast, G = 1 corresponds to one person having all the wealth.

The Gini coefficient for wealth varies from about 0.55 for Japan to 0.85 for Namibia, and is about 0.80 for the United States. Denmark, which has a strong welfare program, has a Gini coefficient of 0.81. The Gini coefficient for disposable income is typically lower, between 0.3 and 0.5, and is about 0.45 in the United States and 0.30 in Denmark.

The meaning of the Gini coefficient can be best understood by looking at Figures 11 and 12, which show plots of the percentage of the total wealth of a population owned by a given percentage of the population, starting from the poorest person. This percentage is an example of a cumulative distribution and is equal to the percentage of the population having wealth less than or equal to a given amount. As this amount is increased, the cumulative wealth distribution must also increase. For perfect wealth equality G = 0, and the cumulative wealth distribution grows linearly with the number of people as we see in the following plot.

*Run Simulation*

Illustration of the Gini coefficient for perfect equality.

**Figure 11:**Illustration of the Gini coefficient for perfect equality.

If wealth is distributed unequally, we obtain a curve for the cumulative wealth instead of a straight line indicating that some people have more wealth than others. If the beginning of the curve is flat, that means many people have very little wealth. The end of the curve on the right must reach 100% of the total wealth in the economy. A very sharp rise near the right of the curve indicates that a few people have most of the wealth.

The Gini coefficient equals the area between the line of equality and the cumulative wealth of the agents divided by the area under the line of equality as shown in the figure below. This plot corresponds to a value of G =0.42 when the tax revenue from a progressive wealth tax is distributed equally to all agents.

*Run Simulation*

Illustration of the Gini coefficient equal to 0.42 found from a progressive wealth tax with equal revenue distribution.

**Figure 12:**Illustration of the Gini coefficient equal to 0.42 found from a progressive wealth tax with equal revenue distribution.

The results of simulations using the models we have discussed show that for proportional distribution of tax revenue, the Gini coefficient is greater than 0.9 for income and sales taxes and about 0.65 for a wealth tax.

If revenue is distributed evenly, the smallest Gini coefficient is 0.42 for the wealth tax, 0.69 for the income tax, and 0.63 for the sales tax. Again reality is somewhere between equal and unequal tax revenue distribution, but closer to the value we obtain for proportional distributions for most countries.