All current political parties in almost all nations claim that the purpose of the government is to serve the people (though they differ on how to achieve that end).
All argue that the inequality they permit is good for the people they govern and offer ideological justifications for inequality
Next: The functionalist justification for inequality
It's worth noting that the virtually universal consensus that the purpose of a government is to serve the people marks a radical departure from most of history and, for the most part, it only dates to the 18th century. Throughout most of history, the purpose of the people was to serve the government. Rulers of states acknowledged little to no obligation to their subjects. Conceptually, states existed to glorify and profit their ruling families. The French king Louis XIV (1638-1715) probably put it most famously: L'État, c'est moi (I am the state). Another good example is a quote often attributed to either Voltaire or Honoré Gabriel Riqueti, Count of Mirabeau: Prussia (in the 18th and 19th centuries) is not a state with an army but an army with a state. That is, the population existed for the benefit of the military, not the military for the benefit of the population. All of this gets at the idea present in the overwhelming majority of state societies throughout history that the obligations of the population to the rulers are very much greater than those of the rulers to the population, which are often virtually nil.
If we conceive of the purpose of the state as making a small number of people as rich and glorious as they can possibly be, then inequality needs no justification: it is the purpose for which the state exists. If, on the other hand, the purpose of the state is to benefit the population of the state, our task is much harder and we must must ask how this benefit is to be achieved. There's no simple answer to the question, but one that is commonly proposed is utilitarianism which argues that the state should pursue the greatest good for the greatest number. If we start there, we would need to ask what distribution of wealth provides the greatest benefit to the most people... It's probably not an accident that this idea comes from the late 18th century, a time when the idea that states were for their people not their rulers was becoming increasingly accepted.