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Individuals care first and foremost to their own interest, and second to fairness. Therefore, most people use the "fairness" argument to their own benefit and care relatively little about what is fair or not.
Some people claim it is "fair" to tax the rich more than the rest of the population, which I am not saying it is necessarily wrong. However, the fact that the world is divided by countries is probably the biggest unfairness of all. The average income of the poorest countries in the world is literally more than 100 times higher than that of the richest countries in the world (Rodrik et al., 2004). Just by looking at their country of birth, one can predict a person's income with highly significant precision (Roemer, 2000). Being born in a rich country is not a skill or merit, it is pure luck. Therefore, it cannot be fair that only a limited amount of people can enjoy such benefits. On the other hand, there are no major movements from citizens of the richest countries in the world trying to finish with their countries' borders and opening their countries to all the poorest citizens of the world. If anything, the trend is in the opposite way. Citizens from rich countries try to restrict entrance to their country as much as possible. In other words, the tax "fairness" argument can be claimed that it used mostly to self-interest rather than to make society as a whole better, which is contraction to the argument itself.
From a tax perspective, there are also great hypocrisies. For example, it makes no sense in terms of fairness to tax citizens who live abroad the whole year and consequently do not enjoy most benefits provided by their home country government. However, regardless of their lack of connections and the time they spend in the US, American citizens continue to be taxed. Not surprisingly, the number of US citizens renouncing their citizenship, i.e. giving up on being Americans, is on its all-time high (United States Treasury). From the best of my knowledge, no country in the world has more citizen renunciations per year than the US. This fact that the US has been taxing people who live and work abroad with wide-acceptance (or at least indifference) from the vast majority of the US population is a great contradiction for people who call for an increase in tax duties on the rich because it is “fair.” If people really cared about tax fairness, they would be making massive protest walks all over the country to take down the US citizenship-based taxation. However, there are no major anti-citizenship-based taxation street movements in US cities. What we see, on the other hand, are major movements to increase the tax on the rich.
Additionally, I also have never seen a large sample study demonstrating that poor people supporting a progressive tax system (i.e. the more you earn the more you pay in tax), continue to support a progressive tax system once they become rich. Thus, it is possible to argue that on average “fair” taxation advocates care for fairness mostly if “fair” makes them better. On average, individuals do not seem to care about "fair" taxation if it does not improve their quality of life. It is hard to defend "fairness" from a moral perspective when the "correct" morality only applies once it benefits you. In a perfect political system, where the rich do not have political advantages over the poor, the fairest tax system is possibly a flat tax (i.e. everyone pays the same amount of tax regardless of the income), as the 2013 Nobel Laureate Eugene Fama claims.
However, the political system is not fair. From a more realist and monetary perspective, it is virtually impossible to tax the rich. Creating high marginal tax rates on the rich encourages migration and capital flight (e.g. Mirrlees, 1982; Piketty and Saez, 2012; Kleven et al, 2013; 2014). The optimal taxation system is probably more toward a regressive system (i.e. the more you earn, the less you pay), such as the Swiss tax system which employs a only small lump-sum tax on the rich.
There are also questions if more money will make all people better off. See "Money and Happiness".