Why inequality matters
Instead, we lefties care about inequality not because we have some idea of what the Gini coefficient or share of the top 1% should be, but because we fear that three things that would make inequalities tolerable are - to some extent - missing.
Firstly, inequalities don't all arise from fair processes.
One condition here - set out in Rawls' difference principle - is that inequality should be associated with "fair equality of opportunity". But this condition is obviously lacking when top jobs go disproportionately to people from the most expensive schools.
Also, many inequalities arise not from free market processes but from what Acemoglu and Robinson call extractive institutions - the ability of the rich to use political power to extract wealth for themselves*. The state, through crony policies such as bank bailouts, generous procurement schemes, corporate welfare policies and intellectual property laws is complicit in this. As David Grusky has pointed out, a lot of inequality is a form of market failure.
Many egalitarians aren't worried about the rising share of the 1% in itself. Instead, we worry about it, when it is combined with falling median incomes - as this is a sign that inequality has become mere predation, rather than a byproduct of a healthy economy. It's the wealth of Emma Harrison that troubles us, not so much that of Wayne Rooney.
Secondly, we fear that inequality has adverse effects. I'm not thinking so much here of its impacts on economic growth, social cohesion (pdf) and other aspects of well-being; the evidence here is convincing if you're prepared to be convinced, and not if you're not. Instead, the danger is that inequality is, as Sean McElwee says, an "affront to democracy." The obvious way in which this happens is that the rich simply buy power. But there are less obvious mechanisms too. One is that the rich's control of business gives them leverage over government policy because - as Michal Kalecki said - they can claim that anything that reduces "confidence" will hurt jobs. Also, as Adam Smith said, we have "a disposition to admire, and almost to worship, the rich." This not only undermines the equality of respect that's part of the democratic spirit, but it leads us to pay too much heed to the interests of the rich and not enough to those of the poor.
Thirdly, we've no great beef with inequality if it is combined with some form of risk-pooling. Even if our first two conditions were met, we'd favour some redistribution to mitigate the effects of bad luck - be it the bad luck of a bad draw in the genetic lottery or of being hurt by a recession. Of course, lefties define luck more widely than righties, but most of the latter should support some redistribution; Hayek, for example, favoured a basic income.
I say all this to make two points.
First, to the right. Contrary to what David Henderson says libertarians should worry about income inequality as it is often the product of statism.
Second, to the left. Getting the rich to pay more tax is nowhere near enough to reduce the inequalities we should tackle.
* Marxists claim that capitalism (and, I'd add, managerialism) are also extractive institutions. We don't need to go that far in this context.