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There are valid moral and economic arguments against high-income inequality (e.g. Stiglitz, 2012). Moreover, assuming that no human race or ethnicity is super to another and a world where everyone has the same opportunities, there should not exist the abysmal economic inequalities that see today both within and across countries. (In this post I will focus on within-country inequality). However, the world we live in is far from equal. In the last 5000 years, no major equalization of income has happened by making the poorer better. A sustainable decrease in income inequality has only been achieved with mass warfare, complete revolutions, state collapse, and catastrophic plagues (Scheidel, 2017). When everybody loses everything, the rich as the ones who lose the most. Therefore, from a realist perspective, should governments even try to decrease inequality?
Assuming politicians' main goal is to make their citizens' lives better, a strong moral claim can be made that governments should at least try to decrease inequality. However, as one of the bases of political science research states, politicians' main goal, on average, is not to improve the lives of their citizens but to stay in power (Geddes, 1995). Politicians may try to improve the lives of their citizens if this helps them to take or maintain power. In other words, helping voters can be the means to the goal and not the goal by itself. This phenomenon can be seen throughout the world. For example, Africa is so poor, to a large extent, because politicians distribute taxes from the poor to the rich who are the ones with real power to maintain or elect a political leader (Bates, 1981). Corrupt politicians, like in Latin America, also leave millions in starvation for their to buy luxury goods and services. American and European governments are also not that far behind (See "The Morality of Legal Tax Avoidance" for a deeper discussion).
Therefore, if decreasing inequality by lifting the poor is virtually impossible and politicians do their best to stay in power, politicians will say they will decrease inequality in case the people who can put or maintain them in power want. But these people, on average, are not economic experts or political specialists. Therefore, politicians have great incentives to give citizens what they think will decrease inequality, like taxing the rich, but in reality, it will not. In the real world, taxing the rich is pretty much impossible. Taxes encourage highly skilled workers to migrate, create major collateral damages, have high implementation and enforcement costs, sometimes generate little revenue, and may make most of the society poorer. Between 1975 and 1986, for example, the US lost revenue trying to tax capital income (Gordon and Slemrod, 1988; Shoven, 1991). Wealth tax has been tried in dozens of countries and has failed miserably in virtually all of them (Boadway et al., 2010). Even if it is possible to lift the poor to make a highly unequal society, like the US or most Latin American countries, look like Europe this would take centuries of hard-fought political battles. And as John Keynes said, "in the long run we are all dead."
To some extent, governments cannot decrease inequality because they are the result of that inequality. Academically-speaking, political institutions and income inequality are endogenous (Acemoglu and Robinson, 2008). Most political leaders in unequal regions are only in power today because they somehow benefited from the system, be it through promising incentives to big firms in exchange of political incentives (like leaving tax loopholes open), spending their own millions in campaigns and buying their way into office, or being born in a rich family (e.g. Graetz and Shapiro, 2005). To some extent, politicians are the system, and to completely remodel the political system to solve inequality would be self-harming. I do not doubt that one or two politicians truly want to completely reform the political system. However, democratic governments have too many vetos to allow a handful of individuals within the system to deeply change it (Tsebelis, 2000). Major political change, such as the ones that can decrease inequality almost always comes from outside the country, i.e. external shocks (Gunitsky, 2014). Domestic factors, like elections, cannot solve income inequality.
In summary, from a realist perspective, politicians should claim that they will solve wealth inequality if the people who have the power to choose or maintain a political leader wants. And that is what politicians usually do. However, governments know that making places like America or Brazil as equal as Europe is not possible in our lifetime. In other words, politicians are self-interested, have incentives to play with voters' irrationalities (like incomplete information on inequality and taxation), and they are endogenous to the unequal system.