Here is a summary of what happens when taxes are added.
Both the sales tax and the income tax lead to almost all the wealth eventually going to one agent if a proportional revenue distribution is used, just as if there was no tax at all. The main difference is that the rich get richer even faster because more of the tax dollars go to the rich.
A progressive wealth tax with a proportional revenue distribution leads to two groups of agents: one group (about 40% of the agents) has almost nothing, and the rest have some wealth. Here is what the wealth distribution looks like.
The wealth distribution with a progressive wealth tax and proportional revenue distribution.
Wealth inequality is reduced if equal revenue distribution is chosen. The poorest agent always has some wealth because every agent receives an equal amount of the tax revenue. For the income tax the wealthiest agent has about 500 times more wealth than the poorest; for the sales tax the ratio is about 300; and for a wealth tax it is only about 50. These ratios would change depending on the tax rates and how often the tax is collected and distributed. Here we show the wealth distributions for equal revenue distributions for the different kinds of taxes.
In the United States the wealthiest person has billions of times more wealth than the very poor, who have little or even negative wealth if they are in debt.
Many people believe that the poor receive more from the government than the rich. However, the results of our simple model suggest the opposite conclusion, because we found that an equal distribution of wealth results in much more equality than exists in the United States.
Consistent with this conclusion is the report by the Corporation for Enterprise Development that “… an analysis of $340 billion in tax subsidies for housing, education, retirement and savings in 2013, [showed that] the top 1 percent received about $95 billion, more than the $90 billion received by the bottom 80 percent combined.” As noted, there are big subsidies to various industries such as agribusiness and oil and gas companies which do not directly benefit the lowest strata of society. The United States is a long way from using the tax system to promote wealth equality or even to prevent excessive wealth inequality.