'Burden' is relative. The person earning £10,000 per year pays £837 in tax and NI.... less than 1% of their income. The person earning £100,000 per year pays £35,391... 35.3%. We are already, therefore, 'asking the richest to pay for the greatest share' and have done for a long, long time.
For the sake of argument, I won't dispute your figures, though I would like to see your source for these amounts...
I think you meant less than 10% of their income for the person earning 10K. A person earning only 10K shouldn't be paying anything in tax at all. They should be receiving aid from the state. Furthermore, the minimum wage should be raised to a 'living wage' whereby they can afford to pay for the cost of living.
The Lib Dems want to abolish income tax under 10K, and on this I agree with them. Not only is this fair, but it will put money in to the hands of the poor who will necessarily spend that money to support themselves.
Raising the threshold to 10K will also benefit everyone else earning an average income, as they will pay slightly less overall as well.
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Secondly, a person earning 10K is subsisting. They are literally struggling to survive. They use all their money on food, rent, bills, and clothes. They often fall short. This is poverty. Paying £837 yearly is a big difference for them. It can mean the difference between eating or not eating, between getting dental treatment or not, between being able to afford the commute to work or not.
Someone earning 100K paying and £35K still earns 65K - which is 2.5x the median (the most common amount). They are still thriving.
I think the have to look at this in context, though. It's not just about the richest paying more, it's about the fact that there are so few of the rich and so many of the poor. Income inequality is staggeringly high in this country. It is now higher than it has been since Victorian times.
I don't think that progressive taxation is a quick fix for this. I don't think you can just keep on taxing the fewer and fewer rich people at higher and higher levels. There is a bigger problem here - so many people are struggling to make ends meet and keep up with the cost of living.
So a more holistic approach needs to be taken which includes reducing prices and the cost of living, and making sure lower earners earn (and keep) more money.