A short story to demonstrate why relative poverty measures have many objectionable features, but the impossibility of achieving them is not one of them.
There was once a fine country with 3 workers named Aaron, Brian and Cathy. They had quite different jobs which paid them quite different incomes. However, their government was worried about the possibility of this leading to one of them being in poverty – they wanted to avoid this at all costs. They defined poverty as the same as the UK currently does as anyone with an income below 60% of the median income.
To clarify, the median is the value you get if you put every single member of the distribution in order from small to large and pick the middle one (or in the case of even numbers conventionally this is rounded up one). It shouldn’t be confused with the mean, which adds everyone’s income up and divides by the number of people.
At first Aaron had an income of £5, Brian of £10 and Cathy of £20. The median income is £10, 60% of this is £6, so Aaron is in poverty and the government didn’t like this at all. So, they decided to do a spot of redistribution (just because it’s the easiest way to show our results: we needn’t necessarily redistribute directly to show this. Market conditions and their relative wages could just change instead). They decided to do this by taxing Cathy’s income at 10% earning them £2 which they gave as a benefit to Aaron because of his low income. Now Aaron has an income of £7, Brian of £10 and Cathy of £18. The median income is still £10, 60% of this is £6, so Aaron is no longer in poverty! In fact no-one is! Yay! We have demonstrated the possibility of abolishing poverty with this measure.
A somewhat more objectionable way to alleviate poverty defined in this way would instead to take £5 of Brian’s income off him and give it to Cathy… (I’m not sure quite what justification I can give to this. Maybe Brian isn’t married while Cathy is and we recognise marriage in the tax system by taxing unmarried people at the somewhat punitive rate of 50%. Meanwhile Cathy receives a non-means tested £5 benefit on account of having a child.) (This is really stretching credulity isn’t it…?)
Now, Aaron (who given my assumptions must surely be married but without children – he really should get on with it if at all possible given the tax incentives) still has an income of £5, Brian is also only left with an income after tax of £5 and Cathy gets £25. Now the median income has fallen to £5, 60% of this is £3, so Aaron is no longer in poverty! A somewhat more cynical and embittered yay!
In our country the distribution of income is somewhat more complex than this, however the simplicity of the number of workers in the country in this story takes nothing away from the mathematical logic. Please note that the taxes are ridiculous, but this doesn’t matter as they have been included for narrative purposes only.