This is the question that I am often asked and will be asked in two days. So I decided to write my answers down.
The argument why inequality should not matter is almost always couched in the following way: if everybody is getting better-off, why should we care if somebody is becoming extremely rich? Perhaps he deserves to be rich—or whatever the case, even if he does not deserve, we need not worry about his wealth. If we do that implies envy and other moral flaws. I have dealt with the misplaced issue of envy here (in response to points made by Martin Feldstein) and here (in response to Harry Frankfurt), and do not want to repeat it. So, let's leave envy out and focus on the reasons why we should be concerned about high inequality.
The reasons can be formally broken down into three groups: instrumental reasons having to do with economic growth, reasons of fairness, and reasons of politics.
The relationship between inequality and economic growth is one of the oldest relationships studied by economists. A very strong presumption was that without high profits there will be no growth, and high profits imply substantial inequality. We find this argument already in Ricardo where profit is the engine of economic growth. We find it also in Keynes and Schumpeter, and then in standard models of economic growth. We find it even in the Soviet industrialization debates. To invest you have to have profits (that is, surplus above subsistence); in a privately-owned economy it means that some people have to be wealthy enough to save and invest, and in a state-directed economy, it means that the state should take all the surplus.
But notice that throughout the argument is not one in favor of inequality as such. If it were, we would not be concerned about the use of the surplus. The argument is about a seemingly paradoxical behavior of the wealthy: they should be sufficiently rich but should not use that money to live well and consume but to invest. This point is quite nicely, and famously, made by Keynes in the opening paragraphs of his "The Economic Consequence of the Peace". For us, it is sufficient to note that this is an argument in favor of inequality provided wealth is not used for private pleasure.
The empirical work conducted in the past twenty years has failed to uncover a positive relationship between inequality and growth. The data were not sufficiently good, especially regarding inequality where the typical measure used was the Gini coefficient which is too aggregate and inert to capture changes in the distribution; also the relationship itself may vary in function of other variables, or the level of development. This has led economists to a cul-de-sac and discouragement so much so that since the late 1990s and early 2000s such empirical literature has almost ceased to be produced. It is reviewed in more detail in the Section 2 of this paper.
More recently, with much better data on income distribution, the argument that inequality and growth are negatively correlated has gained ground. In a joint paper Roy van der Weide and I show this using forty years of US micro data. With better data and somewhat more sophisticated thinking about inequality, the argument becomes much more nuanced: inequality may be good for future incomes of the rich (that is, they become even richer) but it may be bad for future incomes of the poor (that is, they fall further behind). In this dynamic framework, growth rate itself is no longer something homogeneous as indeed it is not in the real life. When we say that the American economy is growing at 3% per year, it simply means that the overall income increased at that rate, it tells us nothing about how much better off, or worse off, individuals at different points of income distribution are getting.
Why would inequality have bad effect on the growth of the lower deciles of the distribution as Roy and I find? Because it leads to low educational (and even health) achievements among the poor who become excluded from meaningful jobs and from meaningful contributions they could make to their own and society's improvement. Excluding a certain group of people from good education, be it because of their insufficient income or gender or race, can never be good for the economy, or at least it can never be preferable to their inclusion.
High inequality which effectively debars some people from full participation translates into an issue of fairness or justice. It does so because it affects inter-generational mobility. People who are relatively poor (which is what high inequality means) are not able, even if they are not poor in an absolute sense, to provide for their children a fraction of benefits, from education and inheritance to social capital, that the rich provide to their offspring. This implies that inequality tends to persist across generations which in turns means that opportunities are vastly different for those at the top of the pyramid and those on the bottom. We have the two factors joining forces here: on the one hand, the negative effect of exclusion on growth that carries over generations (which is our instrumental reason for not liking high inequality), and on the other, lack of equality of opportunity (which is an issue of justice).
High inequality has also political effects. The rich have more political power and they use that political power to promote own interests and to entrench their relative position in the society. This means that all the negative effects due to exclusion and lack of equality of opportunity are reinforced and made permanent (at least, until a big social earthquake destroys them). In order to fight off the advent of such an earthquake, the rich must make themselves safe and unassailable from "conquest". This leads to adversarial politics and destroys social cohesion. Ironically, social instability which then results discourages investments of the rich, that is it undermines the very action that was at the beginning adduced as the key reason why high wealth and inequality may be socially desirable.
We therefore reach the end point where the unfolding of actions that were at the first supposed to produce beneficent outcome destroys by its own logic the original rationale. We have to go back to the beginning and instead of seeing high inequality as promoting investments and growth, we begin to see it, over time, as producing exactly the opposite effects: reducing investments and growth.