Solution? Its a difficult one for the petty left. Take Uncle Blair in Britain. His government did significantly reduce child poverty (down from ridiculously high levels compared to continental Europe), but also allowed income inequalities to further widen
When government subsidizes corporations, bails out banks, prints money, what else do you think is going to happen?
That makes little sense as those aspects of government interventionism are also used in economies with much lower (and sustained) income divides
There are many solutions to the income gap. The key elements which need to be addressed are the following: 1. How can measures be made self-regulating, and thereby more cost-effective? 2. Do measures fulfill a greater economic purpose than eliminating income inequality? In other words, is the measure efficient? Bearing in mind these two inquiries, here are some policies up for debate: 1. A progressive negative income tax system with an added vocational or educational incentive on supplemental pay. 2. Indexing the income tax system to account for non-linear variations in the cost of living. I am certain there are many more valid suggestions.
I've always been a firm supporter of the integration of tax and benefit systems. However, we have to be careful with the efficiency angle. Welfare should be judged on efficiency and effectiveness criteria. Many countries have partial examples of negative income tax systems, but the focus tends to be too much on efficiency